We just went through a tornado warning about as close to us as I ever want to get. Thankfully the bad part of the storm decided to move about a mile north of us.
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If you would like to get caught up on all of the previous installments of this series, please do so by going to the website on which I have been collecting it: PHI—Let’s Get It Right. I’m putting in corrections, and pretty soon navigation aids, there, rather than backtracking on the original blog entries. Some of the sections will be totally rearranged so that the sequence makes sense and some duplications can be eliminated.
This has been a long megaseries, and no one is as surprised by its length as your fatigued bloggist. The initial motivation was a pretty modest one. As I stated right at the outset of this series, my point has been to introduce some cautions into how much use a Christian apologist may make legitimately of the golden ratio, the Fibonacci numbers, and phi (ϕ) in an argument for the reality of God. It has not been my goal to give a full description of the history of phi or of all of its occurrences, let alone the Fibonacci series. Still, I wound up adding a whole lot more material than I had originally intended since I realized in the process that the need for analysis was greater than I had anticipated, and that any number of items would not make sense without providing at least some background.
Having occupied myself with these things for several months now, I’m thinking that the phenomena connected to phi or the golden ratio are best combined with other features of the world for the sake of an argument for God’s existence. Still, the fact that phi shows up in so many different and apparently unrelated places surely is remarkable. As you have read, I am even more amazed by its intrinsic properties as they emerge in geometry and algebraic equations than in their occurrence in nature. Consequently, as a believer in God, I do see his handiwork as an allknowing and allpowerful creator in this aspect of the world he made. When I say “world,” I’m referring to both the physical world of matter and the world of numbers.
However, it is in the much broader context of mathematical and physical reality that the evidential side of the subject matter becomes strong. Phi is a number; in fact, it is a unique and truly remarkable number. So is pi, though in different respects than phi, and so is e, a number whose characteristics would take too long to explain here. Let us not forget 0 (the “additive identity constant”) with her many characteristics, or 1 (the “multiplicative constant”) to whom we did not pay much attention in this discussion. There is i, the square root of 1. There are so many kinds of numbers, as represented by their sets from ℕ through ℤ, ℚ, ℝ, to ℂ, as well as special subcategories, such as ℙ. When we come to terms with the complexity of the universe, while simultaneously recognizing how mathematically finetuned it is, it becomes impossible for me to say, “It just happened."
The evidence for God is there. Unfortunately, some people confuse the creator with his creation and think of the created order as God. However, nothing can be the cause of its own existence and, if there are indications of intentional regularity in the universe, they reflect on the one who made it. When I think of various nontheistic theories of the origin of the universe, I can’t help but wonder how anyone can be satisfied with them. For example, take Michio Kaku’s claim that our present universe is the result of a collision of two previous universes. This idea not only strikes me personally as bizarre (which by itself does not count as evidence against it), but leaves all of us hanging with the question of what caused the existence of those previous universes. Are we back to turtles supported by turtles?
Most important for this series, though, has been my intention to encourage Christians to test the evidence before using it. I have been quite critical of the unbridled way in which many people have attempted to “find” the golden ratio in various parts of nature, architecture, and art. For many of them, the need to find phi in virtually every beautiful building or painting, seems to be driven by a mystical outlook that directly links beauty and the “golden number.” As far as I can tell, a direct causal relationship between phi and the perception of beauty has neither been completely undermined nor proven. However, there is definitely no necessary relationship between beautiful objects and numerical patterns embodied in them. In many cases where the Fibonacci numbers appear, e.g., among flower petals, their purpose appears to be practical rather than aesthetic. This assessment does not take away from its wonder or from its contribution to the evidence for a Creator, but it does extend our understanding of phi beyond creating visual appeal.
Phi is a number. More specifically, it is a number that arose out of the relationship between various geometric lines found in the pentagon. It belongs to geometry, analysis, and number theory. It is not something that we merely run across in our daytoday existence.
Consider the example of the bones of a finger, which we mentioned earlier under the heading of “Phi in Nature, part 3.” Why are we interested in their relative lengths? If, indeed, they do measure 2, 3, 5, and 8 units in length, that discovery per se doesn’t tell us much. 8 divided by 5 equals exactly 8/5 a nicely rational fraction of two integers, which can also be expressed as 1 3/5 or 1.6 with no further remainder. There doesn’t seem to be anything special about it in isolation. If we were ignorant of any further implications, such as the Fibonacci numbers or phi, we might write down the results of our measurements, but we would probably not excitedly post them on a website to share with the world.
The reason why we are intrigued by the lengths of the finger bones is because we already know that these numbers have a larger significance: they are a part of the Fibonacci series, which converges to phi. We fuss over these numbers because we anticipate what’s coming. When we do nothing but measure the proportion between the proximate phalanx and the metacarpal, there is a huge difference between the result we get,1.6, and the number we would eventually reach if we could go on forever with the measurements, namely ϕ. We do not see ϕ exemplified on the xray, and our limited data would not allow us to compute ϕ. However, we have learned previously that the Fibonacci series will converge to phi (1.618033…). Thus, we find the Fibonacci numbers in nature and give them significance based on the Fibonacci series, but the series is actually a mathematical entity. It is the outworking of a recursive equation,
F_{n} = F_{n1} + F_{n2}
with startup values F_{1}=1 and F_{2}=1,
not a principle emerging from the observation of human digits or fecund lagomorphs.
So, what am I saying? Nothing that I haven’t said many, many times, but ever more frequently over the last few years. Apologetics is not about memorizing answers and arguments. We can leave that approach to the internet atheists and their irrepressible need for rabbit trailing. It’s about finding answers to relevant questions as a part of the total project of demonstrating the truth of Christianity, both in evangelism and in dealing with our own doubts. And in that context, memorized answers (“If they say this, then you say that”) without a personal understanding of both the question and the answer, are of very little help, if any.
I’m afraid that phi is a case in point. It’s easy to list it as one of the marvelous aspects of the world that points to the Creator. But I need to ask, to what extent can you substantiate the makeup and meaning of the golden ratio? Can you discriminate between what is science and what is pseudoscience? What is nature and what is numerology? What is truth and what is fantasy or deception? I’m hoping that this series has helped create a little more understanding and, perhaps, even a little bit of interest to delve further into this topic or the role of math in apologetics.
As you have seen, I have built bridges by means of hyperlinks into some of the blog entries and even more into the combined site so that you can jump over some of the meanestlooking equations and calculations. It’s a great feeling when you learn to work through an equation and actually get the result you’re supposed to, but doing the math in all of its fine points if not obligatory. Nonetheless, I do want to say that anyone making use of the remarkable number phi should have some basic understanding of:
1. the difference between phi and the Fibonacci series;
2. the fact that phi is not derived from the Fibonacci series, but that the Fibonacci series converges to phi;
3. the nature of the proportion (whole to large segment = large segment to small segment) that constitutes the golden ratio;
4. the fact that phi originated in geometry;
5. the basic nature of a golden triangle, rectangle, and spiral;
6. some of the genuine occurrences of the Fibonacci numbers in nature, art, and architecture;
7. the lamentable fact that people do fudge the data in their apparent eagerness to find the golden ratio everywhere;
8. our obligation as Christians to be honest and forthright in our learning and speaking. It’s not right to use bad information, even if the person to who you are talking believes it. (Nor, for that matter, is it okay to make up something on the spur of the moment if you don’t know the answer, not even if you think you just experienced a moment of revelatory inspiration [or inspired revelation].)
9. my concern that in order to become a good Christian apologist, one should study fields of knowledge as a whole rather than just picking supposedly good apologetic tidbits out of them.
10. the happiness that results from studying and learning!
It’s August. The temperatures are cooling off, and the days are getting to be rather nice instead of sticky. June is hanging in healthwise, but not doing as well as she should. Let me just relate the minimal story, if you don’t mind. She is showing some serious nutritional deficiencies, which means in her case that the body does not easily absorb them even when their present. The condition may be resolved with megadoses, but the question for us is also what the necessary supplements will do to her digestion. (As for me, I’m pretty good for the most part.)
We had a rather active weekend, by our standards. On Saturday we spent most of the day at a nearby resort/camp ground, though not to return to nature (as much as we want to do that again, given the next opportunity). We were a part of a gospel sing on behalf of MDA, the Muscular Dystrophy Association, which was held in a big “commons room.” It was good to get together again with some of my friends from “Cowboy Church.” There were a number of bands and some soloists. The quality was variable, of course, though nobody was really bad. I did one set in the afternoon, right after which I accompanied Susan and Bob, former fellow members of the Tippyditch Singers and Cowboy Church performers, on my bass.
It was my first time playing the bass publicly in well over a year, as well as finally using the new amplifier I received Christmas before last. So, that song was my warmup, as it were. Then, later on in the evening, I played bass along with another friend. By then I felt quite comfortable on the instrument again and I confess that I surprised him a little bit when, out of sheer high spirits, I inserted some lead lines.
Then yesterday was Jellypalloza, the celebration of the fourth anniversary of StreetJelly.com, one of those days on which all regular StreetJelly artists are invited to do a short set. I did mine in the early afternoon, wearing the recommended tiedie look. My program consisted of a more or less random selection of songs I like in the folk, country, and gospel categories. This coming Thursday night is the fourth Thursday of the month, which I usually designate as an allgospel night. My customary time is 9 pm Eastern, and—as always—it will be live on the internet at StreetJelly.com.
Last night we got together with the “boys” and their wives at Sitara Indian restaurant in Muncie to celebrate my birthday. I have totally fallen in love with their lamb biryani, which they prepare Hyderabad style.
Where else in nature can you find the golden ratio?
One suggestion, advanced by “Mr. Phi,” Gary B. Meisner, is in the proportion of bones in our arms and hands. I have referred to some of Meisner's expositions in other contexts several times already. He is definitely a phi advocate, but a critical and honest one.
Let us look at a couple of instances. I have approached them with as much skepticism as I could reasonably muster. Still, by the time we're done, it looks as though the proposed proportions are present, at least in acceptable approximations.
First, here is Meisner’s depiction of an arm with a hand:
The golden ratio in the human arm and hand,
picture by Gary Meisner from his site.
I went through my usual process of measuring the pixels of the various lines, and here is what I found. Remember, that the dimension of the picture depend on your screen size, magnification, etc. But the proportions should hold true within technically possible parameters.
Section  Length  Proportion 
Forearm  216 pixels   
Hand  136 pixels  Forearm to hand: 1.59 to 1 
Forearm + hand  352 pixels  Forearm + hand to forearm: 1.63 to 1 
Those numbers, i.e. 1.59 to 1 and 1.63 to 1, are certainly quite close within the inevitable margin of error, allowing for fallibility in my measurements as well.
I’m not sure about the placement of the white line separating the hand from the arm. It seems to me that it may be a little too far to the right, past the wrist and cutting off a little bit of what I would consider to be hand rather than arm, and if we moved it a little further left, the ratio would go down. Measuring my own arm and hand did not come up with similar results, but pretty much stuck to 1.5 to 1 even with the dividing point where Meisner put it. I'm not saying that Meisner fudged, but there does seem to be a disparity in my judgment vs. his.
Then there is the proportion of the various little bones of the hand and fingers to each other. We are looking at four components for each finger: three phalanges and one metacarpal, all of them rooted are in the set of carpals.
Diagram of the bones of the hand
by Mariana Ruiz Villarreal (LadyofHats)
Wikipedia Public Domain
Here is Meisner's xray view of one forefinger.
The bones of a human forefinger.
Picture by Gary Meisner from his site
As you can see in the picture, the size of the little bones follow the Fibonacci numbers more or less. Let's agree on more than less. I’ve copied the picture and sunk the lines from the ruler down across the pieces, so as to in order to enhance our ability to visualize the approximations.
It appears that, given a bit of leniency, the Fibonaccis are there: 2, 3, 5, 8.
Still, I do have a little bit of a problem with this display. I don’t have an xray of my fingers handy, and I can only measure my own finger externally. But that fact doesn’t mean that I have no way to test what Meisner is saying. The fingernail is supposed to represent one unit of length. I can’t measure the bone underneath my flesh and skin, but I can measure how many fingernails long the distal phalanx is, and that comes out pretty closely to two units, by assumption and some observation. However, measuring the second section (the intermediate phalanx) does not get me close to three by that scale. At first glance distal and intermediate bones seem to be equal; depending on how I position the finger and ruler, I can get it up to 2.5 fingernails long, but that’s all. Thus, if you use my finger as in any way representative and allow for external measurements, the Fibonacci series is not quite as prominent as one would like. Then again, my fingers may be oddly proportioned in various ways. (Speaking of which, I’m happily looking forward to my trigger finger surgery, still a couple of weeks or so away.)
The lesson is simple, but important. The regularities of nature, even when they manifest a mathematical formula, do not necessarily conform to it with complete mathematical precision. God did not create assembly line robots, but individuals. Theoretically, one could say that each individual's departure from the mathematical template is a flaw or defect. But it's the departures from the theoretical blue plan that makes us individuals, unique, and worth knowing.
It's still early August, but the local schools are already in session. The weather is hot and humid; I wonder if those kids are learning anything. Or, let's say, if they're learning more than under the traditional calendar when schools used to start after Labor Day.
June and I haven't been doing a whole lot right now after coming back from visiting Ralph and Lisa in South Carolina. What a nice visit that was! The StreetJelly concert was a super highlight, but it was good all around.
Speaking of StreetJelly, I'll be doing my "regular" show tonight (Thursday) 9 pm EDT. In fact, that's only about an hour from now, so I better hurry up. Back to just me, Sarah (the guitar), maybe some other instruments, and you, my faithful viewers and remote backup band. The theme is the generic "Summer Sunshine," and I'm bringing out some songs I haven't done in a while. Please join me and help me have fun!
And furthermore, speaking of playing the guitar, my painful "trigger finger" has returned. It showed up last year somewhere around this time, and at the subsequent doctor's visit, an injection of cortisone did the trickfor about 10 months. So, at a visit yesterday the doctor offered to let me get more shots at decreasing intervals one after the other ... not a good thing ... or get surgery. Actually last year, I had expected surgery, and the good results of the shot were a real surprise, but, obviously, that's not the kind of thing someone would just want to maintain over and over again. So, I'm scheduled to have the problem fixed on September 9, one day after the anniversary of my day of amnesia. It'll take a couple of weeks to get back to normal, and it's supposed to remain that way then. We'll see what happens with my musical efforts during that time.
The summer Olympics are well underway, of course, with all their usual flamboyance, real and contrived drama, and some aweinspiring accomplishments. It's the one time every four years that we all get interested in gymnastics, and the women's team has definitely made it worthwhile. Congratulations to the "Final Five" for an entertaining gold medal performance.
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Well, here's another occurrence of phi in nature: the genealogy of a male bee, commonly called a drone. We get to it by way of the Fibonacci series.
First, some basic facts.
1. Your basic worker bee is a female bee descended from an ovum supplied by the queen bee and fertilized by a drone. She collects the nectar in blossoms to make honey, distributes pollen among flowers in the process, and stings people if they deserve it from her point of view. Worker bees have no role to play in reproduction other than the coronation of a new queen if one is needed.
2. A queen bee is a female bee who has been fed large amounts of "royal jelly," a product that turns her into a reproductive machine. She is the daughter of a previous queen bee and a drone.
3. I've mentioned the drones above as the male bees who fertilize the eggs produced by the queen bee. That's what drones do. In fact, it's the only thing that drones do. The unique feature of a drone is that he does not have father, but only a mother, namely the queen bee. Consequently, he only has one set of chromosomes, as opposite to the normal two. Scientifically, it's called "haploid," a term that refers to having only half of the usual numbers of chromosomes (in contrast to the more usual "diploid").
On the right is a diagram of the parentage of a working bee. Father drone and mother queen give birth to a girlbee, who will probably never become a queen. If for some reason, she were to be chosen to become the next queen, her sisters and halfsisters will feed her large amount of "royal jelly." Her abdomen would enlarge, and her ovaries would morph from being useless vestiges to high capacity organs. Otherwise, she's condemned to a life of celibacy and hard work. All humor aside, the queen probably works harder than all of the other bees, spending her life giving birth to one larva after another, and it doesn't look like much fun.
So, let's look at the ancestry of a drone. In the diagram below, you can see that every drone had one queen as parent, and every queen had two parents, a drone and another queen. Common female worker bees have no place in this depiction because they do not actually participate in the reproductive process. And, needless to say since you can see it yourself, the number of ancestors of the drone follows the Fibonacci numbers as we go back generation by generation. Thereby, they are headed ultimately to converge at phi.
This entry comes to you from deep in the American South, where we are visiting brother Ralph and his wife Lisa for a few days. This has been a totally spur of the moment trip, the kind of thing you can do once you’re retired and the nest is empty. June is continuing to struggle a bit healthwise, and we’re looking forward to her appointment with Dr. B on August 17. Speaking of that date, you may also recall that it’s the day on which we celebrate the birthdays of William Carey, Davy Crockett, and some other really cool people.
We’re having a great time together here. In case you didn’t see the post on FB, on Thursday night Ralph and I sang and played together again for the first time in quite a few years. There was a time when neither one of us thought we would ever perform again as individuals, let alone as a duo. So, last night's StreetJelly.com was an incredible blessing for both of us. We had a total riot, and I think that many folks in the audience had a good time, too. There’s a video of us rehearsing on my Facebook timeline.
Now I'd like to come back to the topic that I may have treated too cavalierly earlier, namely the appearance of the golden ratio and ϕ in nature.
Phi in Nature
Do I need to define “nature”? I would think not, but it may be helpful if I emphasize a few of its characteristics. Somehow I want to maintain the idea of “nature” as that part of the created order that has not been formed by human beings. Of course human beings are themselves a part of nature in important ways. So, maybe rather than saying “formed,” a better word might be “subdued” or perhaps “governed.” A sunflower is a part of nature; a picture of a sunflower, strictly speaking, is not. Nature includes our bodies (to a large extent anyway) as well animals, plants, germs, diseases, rocks, astronomical objects, subatomic particles, and lots more stuff. Nature manifests signs of being made by an intelligent personal being, though nature seen as a whole is neither personal nor intelligent. It contains personal intelligent beings, but nature per se cannot think, plan, wish, imagine, invent, like or dislike. For example, contrary to the popular saying, nature does not “abhor” a vacuum. The regularities that we see in nature lead us to conclude that a vacuum will be filled with air or some other gas as soon as soon as access is provided. However, nature has no opinion on this issue—nor on any other, for that matter.
A recent article in the Smithsonian (Danny Lewis, “Why the Turtle Grew a Shell—It’s More than Safety”) introduces us to some recently discovered ancient turtle fossils. These newly available specimens shed fresh light on the reason why turtles may have developed shells. It was not for the sake of defense against other animals, as we might think, but as an aid for them to burrow underground in order to survive the inhospitably hot climate of South Africa, their abode at the time. The article closes on a cautious note: "While more research needs to be done to determine whether the earliest turtles known to have shells were diggers themselves, it just goes to show how adaptable nature can be." Well, I’m afraid I need to add one more item to my list above: nature cannot be “adaptable” either. The term implies foresight and intentional reactions; but the impersonal forces of nature cannot qualify as being “adaptable” in a meaningful sense. You can say that a certain life form became adapted thanks to the potentials found in nature or that some species have turned out to be more adaptable than others in hindsight. However, any kind of intentionality on the part of “nature” or the species in question is overstepping the boundaries of evolutionary biology—unless you accept that there is a Creator and Sustainer who has been supervising the entire process. If you see something “smart” in nature, it’s not nature itself that’s smart because nature itself has no mind. The smartness points us back to the one who made nature. 
Having said that, there are some wonderful examples of apparent smartness in that collection of things in nature, and, unsurprisingly, some of them are classified along that line due to the presence of the golden ratio.
How can we know if some item or phenomenon exhibits ϕ?
1. For some things we can take measurements to see if there are lines that are related to each other in the golden ratio. This is an easy method for human architecture and pieces of art, though we also have a large amount of room for fudging there. It’s also possible to do so for some natural items, such as crystals. For the most part, though, it’s not a practical way to proceed.
2. Bring on the Fibonacci numbers. It appears that some things come only in numbers of the Fibonacci series. Furthermore, if we can find relationships based on the Fibonacci numbers in nature, we can avail ourselves of the fact that they do converge to ϕ, and so the series in question can be said to be in keeping with the golden ratio.
3. Measure the angles. This approach is particularly applicable to anything that appears in spirals. Remember that we said that logarithmic spirals, of which the golden spiral is a special case, distinguish themselves from Archimedean spirals by the fact that their increase in length maintains a consistent ratio, so that the spiral moves away from its point of origin at a much faster rate. In that case, any line drawn from any segment of the origin of the spiral should manifest the same angle.
Thus, in the graph above, the angle at which any of the three lines intersect the spiral is the same. Livio helps us calculate the value of that angle for the golden spiral.
360° ÷ ϕ ≈ 222.5°
Because this result crosses the 180° line, we should come to it from the other direction.
360°  222.5° = 137.5°
This is the angle at which all the lines cut the spiral whenever they intersect it. If the origin of the spiral converges to the center of the graph (i.e. point 0,0), then the x and y axes will also intersect with the spiral at that angle.
Please keep in mind that a golden spiral is a logarithmic spiral, but not every logarithmic spiral is a golden one. Nevertheless, Livio makes the case that golden spirals are very likely the ones in nature. Apparently, in whatever way living beings expend energy in assuming formations, the golden spiral based on phi is the least demanding. So, when we see a logarithmic spiral in nature, the poor chambered nautilus notwithstanding, there’s a good chance it may also be a golden spiral.
Let me try to explain the above point a little more. Livio characterizes phi as the “most irrational” among the many wellknown irrational numbers. Let’s make a quick comparison between ϕ and π. In some ways, ϕ is more approachable than π; in others it isn’t.
, :

Pi 
Phi 
Value 
3.141592… 
1.618033… 
Can it be derived from an algebraic formula? 
No. π is “transcendental.” 
Yes. ½ (1+√5) = ϕ 
Can it be approximated by a fairly simple rational fraction? 
Yes. 
No. 
Calculating ϕ involves referring to another irrational number, namely the square root of 5. The approximate value of one irrational number is computed by including another irrational number. Consequently, if any items are arranged in a way that makes use of phi, no two items are ever going to be sharing one or the other coordinates in space.
[A similar thing applies to time. On occasion one hears the idea that, given an infinite amount of time, it becomes, not only probable, but even certain that every event of the present or past will recur at some point in the future. But that’s not true, as can easily be demonstrated with the following mind experiment conceived by Georg Simmel (18581918) and reported by Walter Kaufmann (19211980) in his magisterial book, Nietzsche: Philosopher, Psychologist, Antichrist. (Fourth Edition, Princeton University Press, 1974, 327). It uses pi, but phi would work just as well in light of the above considerations. Think of three disks rotating at different speeds. One completes a full rotation every minute. The second one does so every 2 minutes (thus, running at half the speed). The third one has a cycle of 1/π minutes.
The disks start at a point we can call zero (maybe the twelve on a clock face). That particular arrangement will never occur again. Similar events, though not formally devised by anyone, occur frequently since pi as well as other irrational numbers come up over and over again in everyday life.]
So, let’s look at a particular example now and a few others next time.
Flower petals and plants
1. It is an observable fact that the number of petals on many species of flowers come in Fibonacci numbers. Once again, I have noticed that some of the same pictures appear website to website, and I cannot credit the person or organization that created them in the first place. Here is a photo with a set of pictures that demonstrate the “preference” of many flowers to take the numbers of their petals right out of the Fibonacci series.
2. Even more interesting is the phenomenon of how the petals of many flowers are arranged. Petals and leaves frequently form a logarithmic spiral around their center. Here's why this pattern is a valuable asset for a plant.
Imagine a flower with several rings of petals, and it takes exactly four petals to make up one ring. So, we start out with a nice flower with four petals surrounding the center.
But we stipulated a flower with several rings of petals, and the second ring also follows the rule of four to a ring.
Clearly, this isn’t going to work too well because the second layer will just cover up the first. And if we continue that pattern, the third layer will cover up the first and second.
The flower is in trouble. How might it work better?
The flower would be far more functional if the relationship among its petals was not governed by such an easy integer as four. But, if we stay with straightforward rational pattern, the same thing will happen. Increasing the numbers of leaves by decreasing the ratio with larger integers, we would just get more layers of petals or leaves pancaked together.
We could play out this scenario further, but you already know where I’m heading with this. In many plants, the petals are arranged according to the golden angle of ca. 137.5°, and thus form a golden spiral. Thereby exposure to sun and rain, the attraction of insects for pollination, or whatever else the plant needs to thrive is maximized. Rather than my pitiful drawing above, we can see a beautiful rose that incorporates a rather complex as well as beneficial mathematical pattern in its structure.
Is this amazing? I think so. Does it point to the idea that a very knowledgeable and powerful being must have overseen the creation and further adaptation of flowers? I should think so.
There is a point that someone could make and unreflectively think that it counted against the above idea. It could be, as Livio hypothesizes, that the Fibonacci sequence and formations according to ϕ represent the least expenditure of molecular energy in the formation and continuation of plants. That’s a good idea, and I already mentioned it above, and—in fact—it is a great idea. It so great that we’ve only pushed the issue one step further back. It only increases the wonder of the phenomenon, and, thus, leads us to an even stronger conclusion that the principle was installed by a supreme creator.
More examples next time.
It's cooled off a bit compared to last week. So, I got a little bit of gardening done yesterday. Also, the swimming pool of Smalltown, USA, is open once again. It was closed all last week because it is located right along the fair grounds, and last week was the 4H county fair. Neither June nor I were particularly interested in the fair this year, partially because of the heat.
I paid a visit to Dr. M, my skin doctor, today. I expected him to scold me for not wearing long sleeves all of the time or not using sun block enough. He would have been only partially right, had he done so. As it was, he skipped that part and gave me a stronger cream to put on my arms.  No, I'm going to post any pictures dealing with my skin.
Well, I didn’t get much of a response to my goldenspiraled swan, not that I expected to cause a buzz with it. I had two purposes in posting it. One was to see how one can make things look as though they incorporate ϕ, even for a moment, until one takes a closer look. The poor swan in the picture has absolutely nothing to do with the golden ratio, at least to the best of my knowledge and analysis. With a little adjustment of the picture, I got it to the point where I could make the spiral center focus on his head, and then the body fit pretty nicely into the back, actually. The obvious problem is that much of the left side of that picture contains nothing but water. So, the total dimensions for the picture notwithstanding, the swan has his own proportions, which don't come out anywhere near the golden one, once you let the water drain out. The other purpose I will tell you about some other day.
Maybe.
I am working through and improving the entries in this series on its separate site. In the process I am not correcting the previous blog posts, unless I run across something so utterly egregious that I don't ever want posterity to see it. Please use the omnibus site to catch up or refer back to.
I have really honestly wanted to get to the Fibonacci numbers and ϕ in nature, but as I keep sorting through material on the golden ratio on the internet, I keep running across its applications in art and architecture that really need some commentary. Where and how the "golden magic" is applied is almost scary. It seems as though there is nothing that we might consider to be beautiful that has not been given the ϕ treatment by goldenratioenthusiasts (hereafter: GREs).
Golden ratio obsession (as opposed to its simple recognition and use) can be traced to to an Italian mathematician, Fra Luca Pacioli (14471517). (See Livio, 12837, for more on this and the next few paragraphs.) Up until then, the ratio had been called by its original name, "the proportion between mean and extreme." Pacioli wrote a 3volume work on it, entitled, De divina poportione  On the Divine Proportion. He saw multiple spiritual meanings in it as something that was both uniquely created by God and expressive of God's nature. However, he did not go so far as to see the golden ratio in every nook and cranny.
Pacioli and Leonardo da Vinci were brought together by fate and politics, and Pacioli taught Leonardo about the "divine proportion" insofar as the latter may not have had not had previous knowledge about it. In return, as Livio (133) put it, Pacioli had the "dream illustrator" for De divina poportione since Leonardo provided the pictures of geometric solids and other graphics for the book. Consequently, there is no problem with looking for the golden ratio in some of daVinci's works and being confident about it if one has found it clearly in certain places. Of course, GREs have spared no effort in finding it in all of Leonardo's pictures, multiple times over in certain cases. Sadly, their unsparing vigor winds up concealing genuine substantiations of ϕ underneath the multitude of invented versions.
For example, efforts to include the Mona Lisa in that group suffer from many of the same problems as those we’ve mentioned before with the Parthenon and the Taj Mahal. People appear to begin by assuming that the golden ratio must be there, but then have to find some place where it might actually fit. For this part, I’m going to inscribe my own drawings on the Mona Lisa, based on what I’ve found on the web. That way, I won't pour any more rain on anyone’s parade in particular. Many versions are just copied and pasted from site to site. If you want to see these and others, some of which are just plain bizarre, look for them on Google or your personal search engine of choice.
After looking at numerous contrived examples, it appears to me that Mona Lisa’s face can actually be very nicely circumscribed by a golden rectangle. It does not strike me as an ad hoc imposition.
Fig. 1 Mona Lisa's Face surrounded by a golden rectangle
This picture limits the rectangle to the open face, taking the measurements at the longest and widest extensions. In other words, there is good reason to believe that the dimensions of the rectangle mean something.
One problem is that GREs are not content to find a single likely instance and must build nested rectangles, spiral, triangles, pentagons, pentagrams, and geometric objects Euclid would never have dreamed of that are somehow supposed to contribute to the painting's goldenness. Obviously (at least to you and me) things don’t work as well with such imaginary placements. Consider the picture below. Again, I have redrawn it based on multiple instances of its appearance on various websites.
Fig. 2 Half of Mona Lisa's head decorated with a golden rectangle
What’s up with that? as the saying goes. This rectangle is a little bit larger as a whole than the previous one; it includes the top of the hair. The proportions are correct, but this rectangle doesn’t frame anything on either side. On the left, it loses itself in the countryside; on the right it cuts through her hair, eye and cheek. The reason it is placed there is because it is actually a part of an assemblage of golden rectangles. Thus, we can create a larger golden rectangle by adding a square to the right of the one that's there, though the resulting rectangle also has no clear moorings on the canvas.
Fig. 3 An extended golden rectangle stuck to Mon Lisa's head
We can go on from there, if we wish, and make more rectangles. My point is once again that, regardless of whether one can decorate the picture with one or more lines of golden proportion, if they don’t have a direct connection to the work of art, there doesn’t seem to be much point to it, and it becomes doubtful that the artist intended to create that particular pattern. Thus, Fig. 1, seems to reveal a golden rectangle. Figs. 2 and 3 strike me as highly implausible.
Speaking of Leonardo and Pacioli, the figure that led to Leonardo's later drawing of the "Vitruvian Man" is also described in De divina proportione, and it would be easy to conclude that, therefore, he, too, must obey golden proportions. That just goes to show how easy it is to conclude something wrong, and for that idea to take on a life of its own under the guidance of the GRE's. There is an excellent treatment of this drawing by Takashi Ida of the Nagoya Institute.
Why is this famous drawing called the “Vitruvian” man? It is based on a description of human anatomy by the firstcentury Roman architect named Vitruvius. His description of the human person, endorsed by Leonardo, emphasizes rational proportions and symmetry. It is in a section of the book by Pacioli that explores various kinds of proportions and ratios, not just the "divine" one.
Did I just say “symmetry”? What an interesting thought! Could it be that symmetry also arouses a sense of beauty in us? Vitruvius thought so, echoing an idea propagated by Aristotle and held to some extent by most writers until golden ratio fever set in.
Symmetry also is the appropriate harmony arising out of the details of the work itself: the correspondence of each given detail to the form of the design as a whole. As in the human body, from cubit, foot, palm, inch and other small parts come the symmetric quality of eurhythmy. [Vitruvius, On Architecture, Frank Granger, trans. (Cambridge: Harvard University Press, 1970), 2627; cited in the article "Beauty" in the online Stanford Encyclopedia of Philosophy.
St. Thomas Aquinas left it open as to what the right proportion may be in a given instance,
There are three requirements for beauty. Firstly, integrity or perfection—for if something is impaired it is ugly. Then there is due proportion or consonance. And also clarity: whence things that are brightly colored are called beautiful. [Summa Theologica, vol.I, q. 39, a. 8, cited in Stanford Enc.
We cannot occupy ourselves with such alternatives for the moment because I must press on.
As far as I can make out, whereas Leonardo golden ratio enthusiasm is a longstanding phenomenon, it is only in its early stages for Michelangelo. Apparently it received its big impetus due to a line discovered on the picture of the creation of Adam in the Sistine Chapel ceiling. (Thanks to David O. for calling my attention to this last year.) It has been discovered that a straight line between the edges of the ceiling segment passing through the point where God's and Adam's fingers almost touch turns out to be in the golden ratio, with the dividing point located right between the fingers.
I have no quibble with this discovery. Of course, the prattle along the line that "now we know why we have always liked that picture because the golden ratio was there even though we didn't know it," is silly. I can't imagine that the picture would be any less appealing if it were shorter by a few inches on one side. For that matter, if I'm not mistaken, there are plenty of compartments of the same length where you can't plausibly stick a golden ratio. So, even though they do not carry the same significance as the one with God and Adam, would I really want to say that they are of lesser beauty?
Next time I really hope for sure: ϕ in nature.
Here is a picture of a swan, and I've framed it with the golden rectangle and the golden spiral nested within it. Is this a good example of the golden ratio?
That's all for now. Please let me know what you think, here or on Facebook.
This entry is a continuation of the previous one, as well as being a part of the lengthy series of phi. I might just mention that, in terms of our physical states, June and I are doing okay. We are still waiting to hear what Dr. B is going to do about some of June’s test results of a couple of weeks ago, and waiting gets old pretty quickly, as we all know.
So, once again I escape into the real world of real numbers, exploring the uses and misuses of phi. I mean, what we used to consider the “real” world has become so surreal at the moment that I can’t bring myself to write on it. Besides, there’s no shortage of commentaries on that realm. So I seek shelter in the part of the world that’s not going to change and its Creator. If you’re tired of reading about phi, or never were interested to begin with, I understand. But please don’t repeat the timeworn myths after passing up this opportunity to reflect on the matter under the gentle guidance of your devoted bloggist.
In the previous entry, I focused on one particular set of pictures supposedly showing how the golden ratio shows up in the entryway to the Taj Mahal, but that this case rests on a glaring mistake, which anyone should have been able to catch apart from knowing any math. Actually, if you examine a larger area than the entry gate, the reason for this strange placement of the golden rectangle entry becomes a little clearer, but no less arbitrary. There seems to be an additional desire for bigger and better golden rectangles, and the inconsistency concerning the entryway not only remains, but is actually expanded by some further dubious interpolations.
I’m going back to the broadly circulated set of pictures that I labeled Pictures 1 and 2 last time. Here is Picture 1 by itself, and, in order to make it easier to talk about what’s happening here, I’ve labeled some important junctures of the lines with letters.
Among the various lines, two apparent golden rectangles are created. One can be described by ACFH, and the other one by BDEG.
These two rectangles can only be golden if they overlap. My measurements fall into the levels of tolerance that we cannot help but allow for. Each rectangle starts from the inside of the opposite decorative door frame and goes either left or right to the edge of the building as it is visible in a picture straight onto the front. These extensions are supposed to be squares, and we know that a square added to a golden rectangle creates a new, larger golden rectangle.
In this illustration, the two squares are ABGH and CDEF—together with rectangle BCFG—are thought to make up two new golden rectangles.
The idea of two golden rectangles created by overlap is clever, though it would have required a lot of subtlety on the part of the architects of the Taj Mahal. I can’t say how such a construction would fit in with the supposed aesthetic appeal of phi. Is our vision supposed to shift back and forth, first catching this rectangle, then that one? For all that I know, such may be the theory and, given the initial assumptions, it could be true, I suppose. But it’s also a leap, and I'm not convinced of the assumptions. We’ve already found that the golden rectangle, as placed in in the entryway, compromises the lightcolored decorative frame as a feature of the building (which is then covered up in the alleged closeup shot that I have called Picture 2 in the previous entry). I don’t know which idea came first, the two larger, overlapping golden rectangles or the imposition of the golden ratio on the entryway. Regardless, in either case, the arbitrary choice with regard to the doorway is still a hindrance.
Moreover, we also need to question the geometrical integrity of the two supposed squares. It should be immediately obvious that points A and D have no architectural anchorage whatsoever. They appear above the balustrade cutting through the small turrets at no particular locations of interest, except that they mark the end of the straight line outward from C to A and from B to D. The internet illustration make it just as clear as my depictions. Here is how it looks on one side:
The fact of the matter is that these squares simply do not exist. The front of the building ends before the square is finished and the wall is bent at an angle to form a new facet of the chamfered corner. (I just learned the word “chamfer.” It’s a decoration on what would otherwise be the stark edge of a 90° corner.) In the picture below you see how the top line of the supposed square ADFH does not stay on course heading left from B to I, though one can blame the photographic angle for that apparent anomaly. However, the shift at points I and J is integral to the building itself since the building has an edge there and begins a new facet. The decorations of the Taj include some optical illusions, but this is not one of them. There is no square here, but an edge in threedimensional space, and I don’t think that it would occur to anyone looking at the Taj in real life, rather than as a flat picture, to see a square at these locations. I, for one, didn’t. Consequently, when combined with the contrived rectangle of the entry way, there is still no golden rectangle here. But, as I keep insisting, numbers are beautiful, but beauty does not depend on certain numbers.
Two “Please Don’ts”
(1) I have zeroed in on one particular set of pictures that illustrate the arbitrariness of imposing the golden ratio on the Taj Mahal. Please don’t think that these are the only such instances. For example, what are we to do with the display below? "Dan" of D&O Celtic Jewelry, operators of Shapeways Stores, is very conscientious as he devotes a lot of effort to explain the relationship between the pentagram and the pentagon on the basis of the golden ratio and gets most of it right—except, of course, the notion that phi is derived from the Fibonacci numbers. That one will probably never go away; the Fibonacci numbers are just too much fun. Then he shows this picture (which is just a clickthrough on my part) and states:
“It’s pretty obvious that the distances and lengths of the main features meet the Golden Ratio, which is kind of mindblowing.”
I really only want to correct common mistakes, and I don’t want to hurt good people or make fun of them or do anything else deconstructive. But how can we possibly see—not overlay—a golden rectangle in the Taj Mahal, when its upper quarter or so is mostly in the sky? In what conceivable sense could that vision be “obvious”? There certainly are parts of the building that fit in with the golden spiral as it sits there, but only a long as the total height of the rectangle is exaggerated.
(2) As I said in my first entry on the Taj Mahal, just as we saw with the Parthenon, it appears to be conventional wisdom that, since the Taj Mahal is beautiful, it must incorporate the golden ratio. However, the variations on where people try to find it, frequently by manipulating the data, add to my doubts as to these claims. It troubles me how many people appear to buy into golden ratio aesthetics without really knowing much about it or seeing the obvious inconsistencies. So, I’m writing this series in order to help people see both what’s wonderful and beautiful about phi and what appears to me to be downright cultic at times. Please don’t think that I’m just out to “get” anyone, except perhaps the folks who seek profit by perpetuating a misconception. Basically, I’m just trying to set the record straight. I’m particularly thinking of Christians who wish to use the mathematical beauty of the world to demonstrate the marvelous hand of the God who made it all. There’s no need to overplay this very strong point by an uncritical presentation of the facts.
I promised a couple of entries ago that, if I were to run across the misleading information concerning the Taj Mahal and the golden ratio again, I would point out the website that carried it. As it has turned out, after a little more searching I found that the picture I had in mind is not on just one site, but is being copied from site to site, apparently without anyone looking at it too closely. In fact, it is sometimes paired with a second picture that is clearly inconsistent with the first one. The site I chose to mention comes from "Project Steam," and it is written in a gentle and friendly manner. The author provides a catchy response to the idea that, since phi's decimals extend to infinity, it cannot be applied anywhere as a measurement. If I may quote,
Now, I am not a mathematician; I am an artist. So my reaction to this argument is to shrug and say, “Meh, close enough.” If that sort of blasé attitude offends your mathy sensibilities, you should probably stop reading now.
The math fan inside of me is taken aback; my writer's instinct loves the prose. I compromise and say that, if you're writing about a subject, you still should avoid obvious goofs. I don't see any math errors. However my concern should be as clearly visible to the artist as to the recreational math fan, maybe even more so. As you know, I have been measuring the purported golden rectangles as depicted in various images by pixels. It's supereasy to do so with Paintshop Pro, and I assume similar programs, such as Photoshop, are just as good in that respect. However, in this case, measurements are totally unnecessary. You can see the manipulations without knowing any math. I doubt that the author of the post is the creator of these picture, but still, I don't understand how he could post them without noticing the glitch.
Here are the two pictures in question, already combined into one image on that website so that a discrepancy should be easy to catch.
Picture 1 (PS) Picture 2 (PS)
Picture 3 (mine) Picture 4 (mine)
I had remembered the adjustment correctly when I reported on it. Specifically, I recalled the placement of the golden rectangle at the entrance (Picture 1 and reproduced from memory in Picture 3), which did, indeed, come close enough to the value of phi to satisfy not merely the artist, but the casual math player in me. But then I realized that there was an issue with how the rectangle was arranged, namely with the top being above the lightcolored frame, and the sides inside of it, as highlighted with the teal lines in Picture 4.
And now, if I may, I would like to direct your attention to the closeup as shown in Picture 2 on the top right. In that picture the rectangle encloses the entire outside of that decorative frame. The difference is visually undeniable. My measurements came out to a ratio of 1 : 1.3, not close enough for anyone I would hope. How can one miss the difference in where the lines are drawn? Maybe the artistic author reposted the pictures as he found them wherever he found them and didn't pay much attention because he didn't think it would make any difference. But it does; phi is all about a number, and a different number can't substitute for phi. I don't intend to drag out a lecture on the virtue of precision in whatever you do. "Good enough" may beno, it isgood enough at times. However, if you're illustrating the golden ratio, and the illustrations are not illustrating the golden ratio, what good is what you're doing? The choice here seems to be between either an arbitrary golden rectangle that disregards the architecture and decorations or a rectangle that follows the features of the building, but not the golden ratio.
My point, once again, is simply that goldenratiomania is leading people to find phi all over the world, linking the beauty of a building to a specific number, and imposing it on objects where it is neither present nor needed. I'm not opposed to finding the golden ratio where it is, and if it's contributing to the beauty of something, very well. But if we feel as though we need to find phi in some contorted way in every beautiful structure, we are doing ourselves and the object a disfavor because then we are mechanizing our aesthetic sensibility.
I have had the privilege to visit the Taj Mahal. When you first walk onto the compound you cannot see it because your view is obstructed by a rather high wall, and you're too close to look over it. Then, once you walk through the inner gate, there it is, the Taj Mahal, right in front of you, larger than lifeand incredibly beautiful. No reproduction that you've seen before can really do justice to the magnificent structure now in full view. Does it embody the golden ratio? I can't find it. Is it an unbelievably gorgeous sight? Absolutely. Would it be more beautiful if it did manifest the golden ratio? I don't see how it could be.
The time here in Mirkwood* is slowly coming to an end. It's been a good stay. The weather has been extremely cooperative.
(*I assume you realize that I'm not using real place names.)
The restaurant here attached to the "Prancing Pony" in Bree has gone bonkers. It used to be a basic countrystyle place where you could get a good meal for a decent price. Now they're charging $10 dollars for just a cheese burger. That's like Hilton Hotel prices, and not exactly in line with out budget. So we take food from the local grocery store to our room. But that's not what I wanted to tell you about.
The first thing I've done every morning has been to go to the stable for my day's ride, while June slept in. I've mentioned the names of some of the horses I've ridden in this park: Whiskey, Bailey, Dan, Levy, Chip, and Bonnie are ones that come to mind. I rode Bonnie once a year ago, and she was my first horse on Monday of this week. As you can imagine if you know me, I always try to strike up a friendship with whatever horse I'm on, talk to him or her, and treat them to some of my cowboy songs, hoping I'm not annoying whoever is riding right before or behind me. So, today I was sitting on a bench by the horses' enclosure, just a board nailed to the fence, with my back to the equines, watching the world as I was waiting for a few more participants to show up for the ride. All of a sudden, I felt a horse nestling the back of my neck. I turned around, and there was Bonnie, saying hello to me. We chatted for a few moments, and then she ambled over for her breakfast hay. I mentioned that little serendipity to Sierra, one of the trail guides, and I added that I did not fantasize that Bonnie really remembered me. But Sierra said she very well might have. Either way, it was nice to have a horse be nice to me.
And now back to the miracle of phi and the golden ratio.
As I said in the last entry, we need to come back to the appearance of phi in the natural realm. Earlier on I skipped most of it, except to give a mere mention of a few examples and a couple of websites. However, I didn't go any further into depth with it, eager to do more math and philosophy than to catalog the appearance of the Fibonacci numbers in nature. That's been done and overdone, I mused. However, on second thought, to do justice to some of those occurrences, we can steep ourselves a little bit more in conceptual matters, and so I'm returning to it. This material may eventually wind up towards the middle of the completed website, right along the other relevant material. One of the first things I've done is to get rid of my earlier rather awkward animation of a golden spiral and to substitute one from Wolfram Alpha, which I then subsequently animated.
I'll reproduce it here so that you don't have to shuttle back and forth:
There are two important groups of spirals, mathematically speaking, and most spirals that we encounter fall into one or the other group: Archimedean spirals or logarithmic spirals. The former are named after Archimedes of Syracuse (287212 BC), and one member of this set bears his name, "Archimedes' Spiral." Archimedes was frequently engaged in applied math (e.g. war catapults, discovering the principle of density displacement), but also made some important contributions to the more theoretical side of math. He came up with a breakthrough method of calculating the value of pi. As I'm thinking of him and his work on spirals, I cannot help but think of the occasion of his death when he was killed by a Roman soldier. According to a popular, albeit unreliable story, when the soldier entered his room, he was engaged in studying some geometric figures, and he was supposed to have said, "Do not disturb my circles!" Was he maybe making further advances in his study of spirals?
Logarithmic spirals are associated with Jakob Bernoulli (16551705), who makes another appearance in this series in connection with the "Basel Problem." Bernoulli was so impressed by this kind of spiral that he called it the "miracle spiracle" spira mirabilis, and asked for it to be a decoration on his tombstone. The tombstone artisan clearly had not studied up on the nature of the spiral in question or perhaps did not understand it. He made a "plain old" Archimedian spiral instead. I don't think that Bernoulli cared any longer at that point, but the difference was very important to him during his life time because it illustrated for him the way in which a thing can be changed and yet remain the same. Specifically, he saw in the logarithmic spiral the renewal of the person entering eternity in heaven. And actually, if it hadn't been for that mistake, we probably would not be talking about it as much.
Just a day or two ago I saw on an apologetics website a notice of a new book, which supports the claim that evidence for God can be found throughout nature. I don't remember author or title, and I don't want to embarass him or his publisher or me in case I'm going to be wrong on this point. All I know about this book at the moment is what the notice said and the picture on the cover, which includes an Archimedian spiral. Books of this nature usually include discussions of the Fibonacci numbers and the various golden angles, rectangles, triangles, and spirals. So, as I glanced at the picture, I remembered Han Solo's words, "I have a funny feeling about this."
So, what's the difference? I'll put it in terms that I can understand and spare us the formulas.
1. Archimedean spirals. Imagine a garden hose that lies flat on the ground, neatly arranged in everexpanding circles. The circumference grows with each rotation, but the distance between each individual piece of the hose and its adjoining ones remains the same. The radius from the center increases, and so does length of each arc. Consequently the angle of expansion (slope, derivative) of the arc flattens. Here is a picture of the Archimedes spiral.
2. Logarithmic spirals. Take the same garden hose. Start to coil it up a little. Measure the relationship between the radius and the arc. Now coil the hose a little more and maintain the same ratio. Well, the segment increases, but you're maintaining the same ratio and thus the same angle of expansion (slope, derivative).Then the distance between segments must increase concomitantly. Extend the coil some more. Again, keep the same ratio, but extend the length of hose. Your distance between hose segments will be increasing some more. As you keep going in that fashion, the spiral becomes looser and looser, and that's because it keeps its ratio. If you were by chance to see just one segment, and you had no idea of how large the magnification was, you would not be able to tell the position of that segment relative to the point of origin of the spiral. Here is a picture that's not in perpetual motion.
A logarithm is the flip side of an exponential number. Express the number 100 in exponential terms, using 10 as your base. You will write 10^{2}.
Now you can say that 2 is "the logarithm of 100 to the base 10."
Logarithms are helpful in many ways. They decrease the distance between numbers to a manageable size.
Expressed exponentially 1,000 is 10^{3}, and so its log to the base 10 is 3. A difference of 900 is expressed with just one integer, going from 2 to 3.
10,000 is 10^{4}, and this time it's a gap of 9,000 units that's expressed with an increase of yet a single number, namely its log 4 (base 10).
The pattern goes on in the same way. Furthermore, if you need to multiply two numbers, such as 100 x 10,000, you can take their logs, 2 and 4 respectively, and just add them. Then you can go back to the exponential version and write out the result as 10^{6}, which is a million. Obviously you don't need logarithms for making simple multiplications or divisions for the powers of 10. However, all numbers have a base 10 logarithm, and so a table of logs can help you with slightly more complicated procedures.
Some of us who have lived through more history than other readers may remember the good old slide rule, which was eventually replaced by the pocket calculator, though not for a few years after I had finished my undergrad as a science major (zoology). It was extremely useful and amazingly accurate. I won't go into its design or functionality now, except to show how the display on a number line (and it had several) was based on a logarithmic scale. You've barely begun with the number 3 by the time you get halfway, but the the distance for each number goes down on a logarithmic function, and you get all of the first ten numbers on each bar. 
Actually, I need to tell you that more often than not, it's not quite that simple. Mathematicians and scientists usually prefer to work with a base different from 10, namely a number that goes by the name of e, called such in honor of the magnificent Leonhard Euler, who is considered the greatest mathematician ever by some people. e is also an irrational number, its value is approximately 2.71828..., and I'm not going into its properties any further. I will just say that e holds as many surprises as phi and pi.
The ratio of increase in a logarithmic spiral can vary from spiral to spiral. As we observed already, the chambered nautilus grows according to a logarithmic pattern, but it's not phi. However, we can find logarithmic spirals in many other parts of nature, where they follow the Fibonacci numbers, and, thus the golden ratio and phi.
It's gotten way too late again. More next time
Thanks to everyone who gave me input on the Golden Ratio test. Here is a table of the results as of 11 pm tonight. Voting is now closed.
What does this mean? Not a whole lot. This was anything but a scientific survey.
Nevertheless. taking it for what it is, we do see that, among those who took the trouble to respond, there wasn't any mysterious attraction to the in the golden ratio. Given these reflections, and, if this quiz has any validity at all, we see that we like the proportion in the range of 1.6, but beyond that , it would be really hasty to draw any further conclusion.
"Can you handle the truth?" as somebody said to somebody else in some movie, whose title I forgot. Of the pictures in the chart, the one that is closest to the golden ratio is number 4. The distinctions have to be quite precise since that's what phi is all about. I must confess that if I had been asked which format I considered most attractive, I probably would have selected no. 3.
Also, on a different note, Thanks to everyone who took part in my StreetJelly show tonight. I never take anyone's attendance for granted, and I appreciate you presence. And thanks for the tokens or pins that you send my way Good night everyone!
This blog entry is coming to you from our little hideout south of Smalltown, USA. It's been a good two days so far. Originally, we thought of driving further, but at this time that doesn't make much sense. I get to do two of my favorite things every day: riding horses and swimming. Today I got to ride on Whiskey again, my favorite, as some readers may remember. June has very little energy, and we'll get into that some later time. Right now there are so many folks hurting and in urgent need of prayer, so we'll wait our turn. Does that make sense?
Before getting back to phi and the golden ratio, just a word in general in the context of Christian apologetics. There's a line of argumentation that we all would do well to avoid. I've seen it used by Christians in conversation with people of other religions, and I've just now experienced it myself. The argument runs this way:
Person A: My religion/world view is X.
Person B: If your religion is X, then you also believe Y, and Y is absurd.
Person A: But I don't believe Y.
Person B: But you must believe Y if you're X, and so, either you don't know your own religion, or you hold to an absurd belief.
Person A: But I really don't believe in Y and it doesn't fit with what else I actually believe.
Person B: Too bad. I know that you are obligated to believe Y, and you're a fool for believing such nonsense.
Person A (slaps himself on forehead): Okay, I see. You're right. How could I not accept the obvious nonsense that you say is a necessary a part of my worldview/religion? Thank you for showing me my duty to believe something absurd so that you can take pot shots at me.
Maybe Person A is inconsistent; maybe Person B is uninformed. Either way, it doesn't make much sense to criticize a person for a belief that he or she doesn't hold. This dialogue would be funny if it came from Abbot and Costello. But in the real world, it's not exactly the best approach.
Typical examples from Christians:
Jews are obligated to believe the Old Testament and are required to expect the reinstitution of temple sacrifices.
Hindus and Buddhists must be pantheists if they only understood their religion completely.
From a nonChristian to me:
As an evangelical Christian, I must accept a young earth theory of origins, which I don't. But it doesn't matter that I say I don't; if I were consistent with my belief in the inspiration and inerrancy of the Bible (as defined in a bizarre way that I hadn't I heard of in a long time, if ever), I would believe in the young earth, and, thus, I'm an idiot for believing it.
This foolishness has to stop.
*****
And now back to our contemplation on the golden ratio and phi. The general belief is that the golden ratio in some inexplicable way catalyzes our appreciation of beauty. Certain experiments have supposedly shown that this idea cannot be documented with evidence. I suspect that this matter is similar to a court case where each side brings in their expert witnesses to support their case. We're talking about a very narrow window. 1 : 1 1/2 is too little; 1 : 1 2/3 is too much. So personally, I would be surprised if eventually there were some clear and objective proof for the aesthetic appeal of phi in art, but it's not something that I can be or want to be dogmatic about. The one thing I do want to caution us about is that, as I have insisted all along, there is a lot of beauty in phi, but its appeal, if any, is not due to some supernatural numerological power .
The table below the video contains a picture of a "Cherokee maiden" that I took ...
[Queue up Bob Wills, Merle Haggard, or Asleep at the Wheel!]
... a couple of years ago. Each one of these pictures has a different ratio of its sides; one of them is in the golden ratio. Is there one that strikes you as more beautiful than others? I sure hope so because some of them are pretty badly distorted. Which one do you like best? I shall disclose which one is in the golden ratio at some future time. In the meantime, have fun with it, assuming that you find this kind of thing amusing or interesting.
1  2  3 
4  5  6 
We're almost done with this series. Next time or so, I'll bolster the section on the Fibonacci numbers in nature.
The Golden Ratio in Ancient Architecture
It’s time to come back to the point of this series. Its topic is the beauty of the number that is usually called phi (ϕ). Given all of my stacking and popping, I think a little recap is in order:
Contrary to the things you frequently may read, phi is not derived from the series of Fibonacci numbers, though they do converge to the value of ϕ: 1.61803…. Its origin lies in the geometry of a pentagon from which we can derive a “golden triangle,” which is distinguished by the fact that the ratio of one side to its base, ...
... is equal to the ratio of the side and base combined to a side to the base.
So that:
.
If we want to express this relationship with regard to a straight line, we can say that there is a line, connecting points A and C, running through point B,
and the ratio of BC to AB is the same as the ratio of AC to B, namely the famous phi: 1.61803 ….
I recounted some fascinating properties of phi, and showed you a few interesting features of phi in relationship to the Fibonacci series. After an excursion on some wonders in π, I asserted that, for Christians, the presence of the Fibonacci numbers in the universe and the beauty within the world of numbers itself should lead us, without hesitation, to affirm the wonderful hand of God displayed in his creation. Then I set out on a long excursion on some scientists and mathematicians who see the world of numbers as divine, but, not wanting to acknowledge the God of the Bible, find God in some unexplained and inexplicable manner within his creation itself.
Given the length and seriousness that this series has taken on, I will flesh out the earlier section on the Fibonacci series in nature. For now, I would like to address two questions that seem to go hand in hand.
*****
We find beauty in many work of art. Some works may be more beautiful than others. Many people say that the perception of beauty may in certain cases be due to the fact that artists have incorporated the golden ratio in their creations. Thus we have two considerations to address:
1. Do we actually find the golden ratio in some of the works that are usually cited as examples of manifesting ϕ?
2. Does the presence of the golden ratio actually trigger our response to consider some things as beautiful?
We need to address the first question first because that's inherent in being the first question.
Artists are free to incorporate the golden ratio in their works to their hearts’ content. And if we find beauty in their production, so much the better. A good example is Salvador Dali’s painting, “The Sacrament of the Last Supper.” Its proportions in internet reproductions appear to be pretty close to ϕ, and I can’t be sure how much may have been lost in either trimming or framing. In this case, a tolerance of a couple of millimeters or pixels can be taken allowed.
For the reproduction I have picked out of the many on the web, this one comes out at 1 : 1.58. Dali leaves no doubt about his intentions, seeing that he inscribed his painting with a dodecahedron, whose twelve sides consist of pentagons.
Apparently the two most frequently used illustrations of the golden ratio are the chambered nautilus and the Parthenon, the ancient Greek temple devoted to the virgin Athena, located on the Acropolis. We already mentioned that the nautilus increases his cells in the shape of a logarithmic spiral, though the ratio is not ϕ. Nevertheless, it is frequently used. Even the front cover of Livio’s book, in which he clearly states that the nautilus is not an example of the golden ratio, greets us with a representation of this misunderstood mollusk. You see that authors are frequently less in control of their books than one might imagine.
The Parthenon is often used to illustrates how an architect employed the golden ratio to endow his work with beauty. It appears to be almost a given that you can find the golden rectangle in the Parthenon. But we must ask, where exactly in the Parthenon do we find the golden ratio? People making this claim usually illustrate it, and the lines that are drawn differ from person to person. Precision is important. When it comes to pictures of buildings, we’re not dealing with millimeters, but with much larger entities.
Here's one example from the webpage culturacolectiva.com. It finds not just one, but two golden rectangles in the building, situated next to each other, inscribed by a golden spiral. Note how the smaller one on the left goes down to an arbitrary line on the ground.
We can contrast that depiction with what we see on the site Design by Day™ Everything you need to know about the Golden Ratio in Graphic and Design.
As most of these pictures do, it extrapolates to the top of the pointed façade, which is lost. The bottom goes to somewhere in the rubble at the bottom of the stairs. The two sides drop in alignment with the eaves and connect to the base in nowhere’s land, cutting the platform where the rectangle requires it, but where there is no architectural indication for it.
Similarly, a site on Greek and Roman art, seems to give priority to the rectangle at the expense of architectural detail.
The location of points A and C in this picture are crucial to finding an additional golden triangle, but don’t seem to play a role in the actual construction of the building. Here is another picture on the same site. Note the pronounced leftward shift of the rectangle.
On Pinterest I ran across this little gem:
The angle at the top is a little more obtuse than in some of the other pictures that extended the lines, and so the rectangle does not need to be as tall to meet the proportions. The base line is located above the stairs. The line is straight, but the placement of the columns is not, except for the front two. And again, the location of the two bottom corners is established by the geometry of the golden rectangle, not by any mark in the building.
We can find the following entry in the Nexus Network Journal^{TM}:
If I read the context correctly, this diagram is intended to make the point that there is no single way of inscribing the golden rectangle in the Parthenon. And that observation obviously negates the idea that phi is incorporated into the Parthenon, and that we see its beauty because of the golden ratio.
Finally, Gary Meisner, on his website ϕ = Phi ≈ 1.61803 comes up with yet another placement of the rectangle, cutting off the eaves.
But he expresses some hesitancy about imposing the golden ratio on the temple front as a whole, because it seems to require too many arbitrary decisions. He then seeks for it in parts of the structure, but, even if that should work, it's out of keeping with the conventional belief that the entire facade attracts us with the golden proportion, and so we'll skip that exercise.
I think that my point should be pretty obvious by now: Lots of people agree that the Parthenon receives its beauty at least partially from the golden rectangle. But there is no unanimous agreement where precisely it is located, a fact that at a minimum should make us a little doubtful concerning this idea. I don't think that beauty is entirely in the eye of the beholder, but the golden rectangle might be.
Actually, a couple of weeks ago, when I was thinking about this topic as a future entry, I decided to try my hand at the process without knowing what to expect. I knew what would be coming with the Parthenon, but I could not recollect ever seeing any pictures of the Taj Mahal in connection to the golden ratio. There are many of them; I had just not seen them. Well, I have a few good pictures of the Taj Mahal that I took in 2006, so I started to select various rectangles on the building and measured their lines in pixels to see if I could get close to a proportion of phi anywhere. The obvious place to start was with the entryway, trying to be as careful as I could, but also being open to the possibility that I could make an error in placement. Nothing I tried there worked out, and neither did the windowlike openings.
I finally decided that I could not impose the golden rectangle anywhere on the Taj Mahal without fudging.
Then, in the course of looking for good pictures of the Parthenon, I came across my first Taj Mahal/Golden Ration depiction. I did not write down the URL at the time (or I misplaced it), but I remember exactly what it looked like, particularly because it was indicated in exactly the area I had been searching. My measurements confirmed that here was, indeed, a genuine golden rectangleor so it seemed.
But then a closer look revealed that the person who posted that picture had been creative in placing the rectangle without taking the decorations into account. His or her top line is clearly above the somewhat lighter frame, but then the sides run along its inside. I thinned out the line a little bit compared to what I originally posted last night and highlighted the frame in order to make the obvious fudge more clearer for anyone who might have not seen it. I think I'm just as happy I couldn't find my way back to that particular URL because that person might just be angry with me. Some people ascribe mystical, supernatural powers to phi and the golden ratio, and probably aren't happy when an obvious fudge comes to light. (If I do find it, I will post it, so you don't have to take my word for it. )
The story of phi and the Taj Mahal resembles that of the Parthenon. "Everybody" insists that it's there. "Nobody" can agree where exactly it is, and there is a lot of fudging going on. I'm not saying that there are no instances of finding the golden ratio in art. But I'm skeptical about how strongly it is represented in some of the traditional supposed examples. I have maintained that there is beauty in numbers; I'm not so sure that works of art depend on numerical values for their beauty. More on that next time.
"Agnosticism" was a word invented by T. H. Huxley (182595) to describe his attitude that he did not know whether there is a God. He took the label of the ancient movement called Gnosticism, whose members prided themselves on their esoteric knowledge, and put an a in front of it, taking pride in what he claimed he didn't know. It was also Albert Einstein's favorite label for his religious outlook, as much as I can make out.
I'm going to repeat a few thoughts here that I brought up in No Doubt About It, 8889. There are two (or more) ways of construing the term "agnostic": constructive and destructive.
An atheist who reviewed the book a number of years ago really took offense at my using the terms "benign" and "malignant." I hope that by now he has gotten over it, though the words are not inappropriate. Bottom line: the constructive form of agnosticism can lead to a growth in faith; the destructive one is irrational and can lead to a person's spiritual death.
Constructive agnosticism can be summarized with the statement, "I don't know if God exists." Well, if you don't, then you don't. For many people coming to this realization is the best thing that ever happened to them. Someone may recognize that this issue is a whole lot more important than, say, whether Pluto is a planet, and start to search for an answer. That's a good thing. For some young people who have grown up within a Christian environment, it can even be a necessary step to make the faith with which they have grown up their own.
However, frequently people use the language of "I don't know" when they really mean "you can't know" or "nobody can know." At that point the agnosticism has become destructive and does not really differ from atheism for all practical purposes. Someone claiming this view must assume that all possible ways of knowing about God have been tried and failed, and, furthermore, that he or she has personal knowledge of all of these tests and their outcomes. This is, of course, not possible. That's why I said that destructive agnosticism is irrational. It presumes an omniscience, which has not been granted to any human being.
Here's the third YouTube video, this one is about Albert Einstein's religion.
There also is a separate Wikipedia article on the religious views of Albert Einstein. It does, indeed, take a lengthy article to compile all the relevant information, and I'm just going to give you a few highlights. The video, some internet sites, and biographical accounts in books are my main sources. I'm just using Einstein as an example and not pretending to put forward a full biography.
1. Einstein called himself an agnostic. I appears to me that many times his use of the term was a genuine expression of humility. He never failed to acknowledge his limits and the limits on knowledge that all human beings shareeven when he was simultaneously trespassing them.
2. This humility is one reason why Einstein eschewed atheism and did not have kind words for those who promoted that position. He saw atheism as a destructive world view that robbed people of the transcendence that human beings need.
3. Einstein definitely did not believe in a personal God, including what he thought was the God of the Bible. There was no shortage of people who held that fact against him, as though he had a greater obligation to believe in God than other people. Incredibly, there were prominent Christians who exhorted him that, since he was Jewish, he was giving Judaism a bad name by not accepting the God of the Old Testament. But Einstein, much like Kaku now, was not open to a "God of Intervention." He considered the idea of a personal God, as found in Judaism and Christianity, a childish fantasy used to instill fear in people so as to make them behave. Einstein had some Christian teachings as a part of his early education, and, for all that I know, his teachers at that time may have been using God as a bogey man figure to frighten children into being good. Needless to say (I hope), we're once again looking at a caricature of the biblical God, andas far as I can tell with my very limited researchhe never pursued educating himself further on a more mature understanding of God. Once he was done with it, all that was left was a patronizing smile from his allegedly more rational point of view.
4. Apparently there were several occasions when some media outlet declared that Einstein believed in a personal God, and these reports made him furious.
5. As mentioned in the last entry, Einstein frequently averred a particular preference for the pantheistic God, as described by Spinoza. Let me just reiterate that his admiration of Spinoza's view requires the romantic glasses through which Spinoza was being read starting in the early 19th century. The things that intrigued Einstein about the universe and the possibility of some kind of a deity are the very things that Spinoza reduced to a flat, undifferentiated monism. To be sure, Spinoza's God was rational and orderly, but the beauty and elegance that mesmerized Einstein are not found in Spinoza's Ethics.
6. Just as we saw with Kaku, it is a whole lot easier to compile what Einstein did not believe than what he did believe. As soon as we try to look more closely at the positive side, we see expectations, beginnings, and inconsistencies without resolution. Here are two quotes that I took out of the Wikipedia article. This is the first one:
"God is a mystery. But a comprehensible mystery. I have nothing but awe when I observe the laws of nature. There are not laws without a lawgiver, but how does this lawgiver look? Certainly not like a man magnified."
Einstein is in awe of the laws of nature, an attitude we should applaud. He goes one step further and asserts that these are laws that must have been legislated by a lawgiver. Once again, we're running up against a highly unusual understanding of the laws of nature, at least as it shows up in the phrasing. The laws of nature are not commandments given by God in the way a government makes laws for its citizens. To repeat something I said earlier, the laws of nature are descriptions and perhaps statistical generalizations. Some of them appear to us to be ironclad and unrevisable. They are discovered by people working in the sciences. So, the idea of a prescriptive divine Lawmaker is somewhat odd for the laws of nature, and it was probably not what Einstein was really saying. I think we can agree that what Einstein meant was that the law of nature are such that one is driven to see an intentionality underneath them. But please note also that Einstein overstepped his professed agnosticism in this statement. He ruled out any anthropomorphic understanding of God, but then he must know something about the one who is beyond our knowledge. If he is truly in the dark about God, then he is not in a position to set up rules as to God's true nature.
This point is important because the descriptions of God that Einstein called "anthropomorphisms" are the attributes of God that lead us to understand him as a personal being. From time to time Einstein used anthropomorphic language about God as well.
Der liebe Gott würfelt nicht.
"God does not play with dice."
Does that assertion mean that Einstein pictured God as a person with hands who performs various actions, but refuses to roll a pair of dice. Of course not. Einstein was using an anthropomorphic image to make a greater point about a being that does not literally have hands or could be tempted to entertain himself with dice. And that is the proper method for understanding the Christian notion of a personal God as well. What's sauce for the goose is sauce for the gander. Our language is limited by our earthly environment, and we cannot express what God has revealed to us about himself without using terms from a finite context and applying them to the infinite. (Some of the best philosophers of religion place the language of religion under the technical label of "analogy.") No, God is not a magnified human being, as the atheist Ludwig Feuerbach proclaimed in his Essence of Christianity. But the only way in which we can talk about God is with human language. The alternatives are either to say nothing at all or to utter something meaningless. Einstein did not choose the former option, and his dictates about the nature of God seem short on meaning. One simply cannot declare the ineffable, let alone make up rules for it.
7. We see the same confusion in the second quotation:
A knowledge of the existence of something we cannot penetrate, of the manifestations of the profoundest reason and the most radiant beauty, which are only accessible to our reason in their most elementary forms — it is this knowledge and this emotion that constitute the truly religious attitude; in this sense, and in this alone, I am a deeply religious man.
I am reminded of the controversy of thirty or so years ago with regard to the work of art produced by Andres Serrano. I'm not going to describe it here because the details don't matter. Suffice it to say, it was rightly considered blasphemous and offensive by Christians, particularly Roman Catholics. However, I ran across one review [Art News 89 (April 1990):163], in which the author defended Serrano. Specifically he said that critics were misunderstanding him; he was not an opponent of religion. In fact, he was a deeply spiritual man who was devoted to a religion focused on bodily fluids. That case certainly is crassly different from Einstein and his beliefs, but the same logic appears to be at work. Whatever Einstein declares to be a religion, regardless of whether it violates any reasonable understanding of religion, is a religion.
religion ≡ whatever Einstein declares to be a religion (def)
Einstein confesses to the existence of something that is almost entirely inconceivable and ineffable. That recognition makes him a deeply religious man, he claims. Once more we are allowed only a scanty look, and the picture can't be filled in because there isn't even a sufficient outline. All we have is a vague apperception that overwhelms us as we study the cosmos. If that's sufficient to be called a religion, then Einstein was, indeed, a deeply religious person. I must admit, however, that I think we're looking at something extremely thin here.
So what? someone may respond. If Einstein was content with that description as his "religion," why is that my concern? Why am I picking on Einstein (and on Michio Kaku earlier)?
If you are asking that question, please read that quotation again and take cognizance of the claim that he is describing the "truly religious attitude." Albert Einstein is not just asserting that this recognition of something beyond is a religion in its own right; he is making an exclusive claim for its truth and superiority. Consequently, that quote does challenge us to interact with it. I cannot help but see Einstein asking the world to emulate his own highly underdeveloped notion of spirituality, inspired by reason but beyond rationality.
I'm going to add one more statement that is going to come across as pretty harsh, I'm afraid. One could say that, even taking all of the above into account, we should pay greater attention to what Einstein is saying in the realm of religion and philosophy because he was, after all, such a wise person. At least some of my present readers will remember that in many ways I am a great fan of Einstein, and I'm enamored with his theorieseven though he was wrong at times. It's easy to secondguess his contributions to the Manhattan Project, which for a time broke his lifelong pacifism, and I can't judge him on that in either direction. But whatever else Albert Einstein was, he was not a man of wisdom. He was very intelligent and wellspoken, and it would be absurd for me or anyone else to diminish the contributions he has made. Unfortunately, those attributes alone do not make for wisdom. Wisdom includes the application of knowledge in one's life. An extremely smart person may make some terribly unwise decisions, and a biography of Einstein will disclose to you pretty quickly that he was a man with some serious faults.
Yes, yes, we all have serious faults, but we're not asking to be given an exemption in order to be considered prophets of an irrational mysticism. One other item that Einstein stressed frequently was that the God who legislated the laws of nature did not legislate morality and that morality was a purely human creation. Sadly, his life confirmed that his pseudoreligion did not impact his personal conduct.
Oh, how I wish he had seen the Creator's hand within creation!
Next time: Back to the Golden Ratio!
June and I were both really wound up when we came home last night after the family fireworks. Plus I really wanted to get that previous entry posted. What I thought would take about fifteen minutes took 2 hours, thanks to formatting glitches, so that set my brain twirling. (And once again, I'm back to writing my code on Notepad, and then copying and posting it to Bravenet.) We finally settled down around 4:30 am, but it was several more hours before I actually fell asleep for a few hours. June tells me that at one point during my sleep I started to sing out loud, which woke her up for a while. But she didn't remember (or recognize) the song. We've both been pretty exhausted all day today.
Please be aware of the fact that all of my previous posts in this serious are collected on my site: PhiLet's Get it Right.
Let us move on to a second video in which the eminent theoretical physicist, Michio Kaku, declares his opinions about God.
(1) Let's ignore the gushing interviewer's comments about the "spirituality" of most cosmologists. If I were doing a oneonone interview with someone of Dr. Kaku's stature, I, too, would treat him as the celebrity that he is. She selfedited her first question as she went along, and I think I have been able to assemble the bare bones of it without the asides and qualifications:
Q: "What is your view on life, and where is it? What are we doing when we teleport life?"
A: "Well, if I had an answer to life, I would have an inside track up there." He's pointing straight up.
Response: Kaku's answer has several dimensions (though not ten or eleven). For one thing, this answer may lead us to infer that Kaku holds to a view of God as transcendentbeyond the world and unknowable. In what follows, however, it becomes clear that Kaku's conception of God is of one who is entirely immanent, who is one with the world and its structure. Another observation I must make is that he seems to have no room for the common belief among those who believe in a transcendent God, that he has the ability to reveal himself and that he has done so. As a Christian, I can't say that I have an "inside track" to God, whatever that could mean. But I know his message and his will because he has disclosed himself to us, not in all of his infinity, of course, but in a way that becomes intelligible to us. Furthermore, what I know about him is available to anyone. When I read the Bible, I do get information about the meaning of life, not exhaustively, but enough that I'm not in the total dark about it. Obviously, nonChristians don't see it that way, and there is no need to make a big point out of that. My main objective is to help you understand what I think Kaku is trying to say, and doing so involves exposing the inadequacies of his theological concepts. If some things that he says don't make sense to you, don't blame yourself. Right here he seems to be referring to an alltranscendent, but essentially unknowable, being. But he doesn't stick to that script for very long.
(2) Prof. Kaku continues to expound his views of God. He refers to Einstein and his answer to the question of whether he believes in God. As a physicist one must, of course, be scientific in dealing with that question, and he clearly takes great pleasure in his elitist standing in such matters. The perception he seems to want to create is that he is more rational and informed than your average poorly educated believer in God. Consequently, he avers, we need to have a clear definition of what we mean by God. One cannot help but see the subtext that people who are not scientists do not live up to his supposed strictly rational analysis. There are two ways of thinking of God, he claims. (I'm treating this part of the exchange in three sections: the limit of two options, the God whom he dismisses, and the God whom he believes in.)
Response: In informal logic, we encounter a fallacy known as a false dilemma. One commits this fallacy when one presents an issue as though it had only two options, and one of them is so unacceptable that one must embrace the other one.
Your only choices are A and B. A is wonderful, while B is totally unacceptable.
"Do you want to be a benevolent pacifist and abandon all physical violence, or do you
choose to be the kind of monster who thinks that we should nuke
all the countries that we don't like and not worry about the consequences?
Hopefully, rational people understand that there are many more options between absolute pacifism and extreme warmongerism. That example is obviously not our topic, but simply serves as an illustration of the kind of polemic that Prof. Kaku engages in. To be sure, there may be times in our lives when one may have only two options to choose from, and one of them may be so bad that it really doesn't deserve consideration. Then there is no fallacy. The actual fallacy occurs when one imposes such a scheme on an issue while either ignoring or being ignorant of other options. My paraphrase:
There are two ways of looking at God: One comes with the clout of being held by scientific geniuses,
while the other one is childish and can be dismissed with a gentle chuckle.
If Prof. Kaku only knew what he doesn't know! There are many more ways of understanding God than a laughable cartoon version and his allegedly scientific one. What grates as much as anything is the air of superiority with which Kaku spreads his peacock feathers of ignorance to the world. He, after all, is a scientist, so he is in a far better position to understand the idea of God than the huddled masses. This is pure arrogance, particularly in light of the fact that, when we're all done we'll see that he doesn't even have any coherent understanding of God. Just to mention a few options in the conceptualization of God of which doesn't seem to be aware: personal religious theism, impersonal philosophical theism, trinitarian theism, unitarian theism, personal pantheism, impersonal pantheism, deism, panentheism of various forms (e.g., Hegel's transcendental panentheism or Whitehead's process panentheism), polytheism, henotheism, and others, not to mention combinations of some of them. I am tempted to say that before one proceeds to declare that there are only two options, one ought to have acquainted oneself with all of the options. But I cannot make that statement because undoubtedly there are many more ways of understanding God that I'm not aware ofan observation that underscores my point. But there are many models well accessible to study and learn about, and to present a picture of only two choices, one infantile and the other unintelligible, clearly demonstrates a lack of learning.
Now, one can come up with a good rebuttal to what I just presented. One could say that, regardless of the model of God one proposes, each of them clearly falls into one of two fundamental categories. But that's not what Kaku is doing. He's not giving us two classes of gods into which we can divide all the available models. He is having us choose between two specific models of God, the "God of Intervention" and the "God of Order." And that's why I must say that he is trying to entice us with a false dilemma.
(3)The first option for understanding God is as a God of intervention who answers all of our prayers, who parts the waters, who kills Philistines on our behalf as per our requests. Einstein, Prof. Kaku tells us, had a hard time believing in that kind of God.
Response: So do I. What Einstein and Kaku are dismissing is the "Santa Claus" version of God that maturing Christians should learn to grow out of. It's not the God of the Bible. Yes, the God of the Bible is a personal God who can and does act within the world he has created. But he is not a god out of, say, Greek or Hindu mythology, who changes his mind frequently in order to accommodate our demands so as to continue to be liked by us.
(4) Kaku endorses the idea of a God of order, harmony, beauty, simplicity, and elegance. A look at the physical universe bedazzles us with those qualities. God would not have needed to create the universe in that way. The God whom both Einstein and Kaku are promoting is the "God of Spinoza," a God of order and regularity.
Response: I cannot say whether either of these two men has ever actually studied Baruch "Benedict" Spinoza (1631677) or, if I may be just a little crass, whether they understood what they were reading. To say that "Einstein read Spinoza's Ethics" does not tell us much. I shall skip the virtually mandated biography of Spinoza and give you just a bit of insight about the core of his system. He set up his philosophy in a form that resembled Euclid's Elements, drawing up axioms, theorems, and corollaries, a procedure that obviously holds a certain attraction for these scientists. Conceptually speaking, he began with René Descartes' understanding of substances, namely that substances do not have properties in themselves, but that the properties are added to an underlying substrate, which is the substance. I like to call this the "pin cushion" theory of substances. The properties are the pins that are inserted into the cushion. They can be removed or exchanged, but the underlying cushion, i.e. the true substance remains the same. Spinoza took off from the notion that substances per se have no intrinsic properties. So, we can consider one substance and declare that it is a substance, and that we know so by a direct rational intuition rather than by the properties it displays. Then we can look at a second substance and again recognize the fact that it has no intrinsic properties.
But wait! How do we know that we're not looking at the same substance for a second time? If a substance has no discernible properties that distinguish it from other substances, it is not possible for us to recognize more than one substance. There is only one substance. So then, all diversity of substances, actions, change, motion, etc. are not truly real, but are mere modalities in which the true Substance manifests itself. It's not necessary here to go through a lengthy argument demonstrating the unresolvable contradiction that runs between identifying infinite substance with finite things. The point I'm making is much simpler than that. In Spinoza's philosophy, the details of the universe, i.e. the beauty, harmony, elegance, and order of the world are swallowed up by simplicity. In fact, this simplicity is an allabsorbing oneness. In short, as Kaku presents his picture of God, it is not the God of Spinoza. Isabelle T. mentioned on Facebook that underlying his thought may be the Buddhist concept of sunyata, the allpervasive emptiness of all.That's a good analysis. But still, that's not the God of Einstein or Spinoza, and in Buddhism there is no such thing as a Creator who made a universe that displays beauty, order, simplicity, elegance, and harmony. Either way, Kaku's God is severely illdefined and selfinconsistent.
(5) The attribute of simplicity is of great importance to Kaku. All the equations of physics can be written down on one sheet of paper. Better yet, the fundamental formula of string theory (Kaku's contribution, he is careful to add) is extremely short. The shorter, the more elegant. The more elegant, the more beautiful. Kaku's formula captures the essence of God, it would appear.
Response: Kaku may claim that his belief in this strange God is based on observing the order, beauty, elegance, etc. of the universe. But his conceptualization of God has yet to attain order, let alone beauty, or elegance. One feels as though there is something else that Kaku knows (and maybe Einstein did as well) that is hidden from the rest of us. One wishes for a genuine definition or description; he started out by gloating over the assumption that as a scientist he is dutybound to start out with a clear definition of God [my paraphrase]. But such a definition is not forthcoming. I, for one would be delighted if we could start with comprehensibility, not of God, but of what Kaku is saying. Labeling his deity as the "God of Spinoza" doesn't do it because Spinoza's God is clearly not Kaku's God. Displaying his formula to the world doesn't convey anything to most of us, except that he loves simplicity, elegance, etc. But that's where we started, so that equation doesn't tell us any more about the nature and function of this Godthing. We're pretty clear on what Kaku does not believe, namely a caricature of God as a cosmic Santa Claus. The only thing we can say definitively is that we still don't know what he really means by God and that this God supposedly differs drastically from all other conceptions of God. There seems to be an inside track, after all, though the path does not appear open to most of us.
Happy Fourth of July to all relatives, friends, acquaintances, fans, colleagues and work associates of the past, present, and future. Whether you’re celebrating or not, the day is on your calendar, and I hope it is a good one for you. The question here in Smalltown USA and environs is whether the weather will clear up for fireworks tonight. Just like yesterday, it’s cold (for July), dark, cloudy, and drizzly outside. The hourbyhour forecast has it clearing up by around 9 pm tonight, which would be the time that it started to get dark enough for our incendiary displays. We’ll see.
Late addition: The weatherman hit the proverbial nail on the head. June and I just got back from a wonderful evening with Nick & Meghan and Seth & Amber at the former’s house in the country. N&M had everything ready for hoboes for supper and strawberry shortcake for dessert, and we had some awesome fireworks. Obviously, Sunako was there, since she lives there, and S&A brought Misha, the aging whippet, and Evey, the baby Great Dane.
I have added all of the previous posts relating to this series to the Phi website. Let’s get ourselves caught up on where we are in this long and drawnout series, beset by many a tangent. I’ve stated that a Christian, scientist or not, should see God’s hand in the beauty and order in the world that God has created, and that such beauty should extend to the world of numbers with a beauty all its own. I expressed my disappointment with those Christians who are too eager to maintain standing in the socalled community of scientists to acknowledge that even the question of whether there is scientific evidence of intelligent design in the cosmos is a legitimate one. The subtopic at this point is those scientific luminaries who appear to go in the opposite direction and seem to be finding God in the mathematical structure of the universe itself.
Michio Kaku (1947– ) is known as a successful physicist who invented string theory and some of its subsequent developments, such as super string theory. He is also an adept populariser of physics and mathematics, and, consequently, has become well known; he quite obviously enjoys talking to the world’s population at large about science and himself, not reluctant to express opinions outside of his area of expertise. As promised in the last entry, here is his YouTube video, entitled “Is God a Mathematician?”
I’m going to follow Professor Kaku fairly closely in this round. Don’t worry; I’m not going to challenge him on the validity of string theory. There are three major figures mentioned in this short talk: Newton, Einstein and Kaku. In the actual video he does not address the question whether God is a mathematician, but of what good math actually is.
Note: I have tried several ways of formatting this discussion, and I found it to work best for me if I include my responses and critiques right along with the summaries.
(1) Isaac Newton. Kaku uses Isaac Newton as an example of a time when innovations in mathematics and physics came up together.
(a) Kaku applauds Newton for asking the question: “If an apple falls, does the moon also fall?” He classifies this question as among the greatest that any member of homo sapiens has ever asked in the 6 million years since we departed from the apes.
Response: Along with all other members of our species, Kaku is entitled to his opinion, as long as we are clear that it is nothing more than his opinion. Personally, I have a problem with his evolutionary assumption, and even if I didn’t, I’m pretty sure that I would not include Newton’s concern among the top questions ever asked by humankind. But that’s a matter of perspective.
(b) Kaku’s narrative goes on: Newton’s answer to his own question was that, yes, the moon also does fall (viz. the moon is attracted to the earth by the pull of gravity). However, Newton lacked the mathematical tools to explain this idea. So, he invented calculus in order to make it possible.
Response: Is that how it happened? I don’t think so. Newton described gravity with a famous equation, oftentimes referred to as the Inverse Square Law (ISL), where G is called the gravitational constant.
The force of gravity between two objects is obtained by multiplying the mass of the two objects, dividing that number by the square of the distance between them, and multiplying the result by the gravitational constant.
I do not see any calculus here. This formula was included in Newton’s Principia Mathematica, which came out in 1687, and which did not contain any calculus. His first publication on calculus did not appear until 1693.
(c) Kaku then summarizes Newton’s conclusion: “The moon falls because of the Inverse Square Law. So does the apple.”
Response: I replayed that segment of the video several times in order to make sure I heard correctly, and the closed captions also bore me out.
“The moon falls because of the ISL.”
Scientists don’t usually say such things. The laws of science are descriptions. Often they are statistical generalizations, and frequently they appear so ironclad that there appears to be no way around them. But there is one thing that they never are, namely causal agents. The ISL describes the earth/apple and earth/moon attractions mathematically. It tells us how to calculate the gravitational force, but it does not explain the presence of a gravitational force. Both the moon and apple would still be falling, regardless of whether we had the ISL or not.
Is this just verbal nitpicking? Should I just dismiss Kaku’s phrasing as a momentary and careless slip of the tongue, something that happens to all of us from time to time? Usually I would say so, but in light of the status that Kaku eventually gives to the laws of nature a little later on, I can’t rule out that he looks at the ISL—among others laws of nature—as more than just a description. Also, as I said above, scientists are usually far too careful to use such careless phrasing, especially, I would think, in a video published to the entire world (at least the part covered by YouTube), so I can’t help but see some intentionality in Kaku’s remark.
(2) Albert Einstein. Albert Einstein is Kaku’s example of a case where a physicist was able to draw on a mathematical method that was already available prior to making an innovation in physics.
(a) Kaku tells us that Einstein came up with a new way of looking at gravity. The key to Einstein’s general theory of relativity was that he envisioned space as curved, and gravity is due to that curvature, not a force of attraction. The fact that he remains in a chair, Kaku assures us, is not because gravity pulls him, but because curved space pushes him down.
Response: I will just say that I have never heard it put that way. More commonly the explanation is that mass and energy create curvatures in space, so that my sitting in a chair causes a little protuberance in space into which I have sunk myself. Kaku seems to substitute one mysterious force for another, i.e. pushing instead of pulling, and I’m just a little confused on why he would put it that way. Still, I’m not going to argue with Kaku over matters that are purely physics. Maybe someone else can comment (nicely) on that description.
(b) According to Kaku, Einstein was also in need of a mathematical means to describe his innovation in physics. But he did not have to come up with anything new; he could simply avail himself of “differential calculus,” which already was in place. After all, he reminds us, studying calculus usually begins with the motion of falling objects, i.e. gravity.
Response: The last part of that little summary is true. Among other things, differential calculus analyzes the rate of change in the velocity of an object in motion at a particular time, e.g., the acceleration of a moving object, or even the acceleration of the acceleration. This is one part of calculus that Isaac Newton and Leibniz had already invented, and if Einstein would first have had to learn differential calculus at that point in time, he would not have been much of a physicist. (The other part of calculus is called integral calculus.)
Einstein did not need differential calculus to make his theory of space rigorous. Professor Kaku's statement implies that we begin the study of calculus with differentiation (aka “finding derivatives”), and that would most likely be true for pretty much everyone, I would think. But Einstein’s math goes far, far beyond “differential calculus.” Yes, there is calculus involved, but so are addition, subtraction, multiplication, and division. Surely all of these mathematical procedures become trivial characterizations in connection with Einstein’s work. If we were to try to cover Einstein’s math with short expressions, we might want to say that it was “tensor calculus” for the special theory of relativity and “Riemann field geometry” for the general theory. I don’t understand why Kaku trivialized Einstein’s work with the expression he used.
Could this just be a sloppy choice of words, and am I maybe nitpicking again? Once again, I doubt it. Who, given just a small amount of understanding of what is involved, would say that Einstein resorted to “differential calculus” for his general theory? This is not just careless, but misleading. One of the world’s foremost scientists must know better than to make such a mistake. My hunches: Kaku is trying to put us at ease by using a term that is not going to frighten us away. After all, we’ve all studied calculus, right? Could there be a further reasons to minimize Einstein and, thereby, establish a contrast to himself?
To get a flavor of relativity math—or maybe even to learn it—may I suggest Peter Collier, A Most Incomprehensible Thing: Notes toward a (very) gentle introduction to the mathematics of relativity (Incomprehensible Books, 2012). I must say, though, that Collier and I might just disagree on the meaning of “very gentle.” I suspect that there actually is no gentle way of learning this material.
(3) Michio Kaku. As you can see in the video, Professor Kaku is not exactly humble about his accomplishments, and I, for one, want to congratulate him on his life’s work. Even if string theory, superstring theory, Mtheory, etc. are not going to hold up in the long run (and I am definitely in no position to judge that), his ideas have been significant in keeping the search for ultimate physical reality moving ahead.
(a) The holy grail, so to speak, of theoretical physics is a unified field theory, an explanation for everything. Einstein spent the majority of his life in a futile quest for it. Kaku believes that he has found it with his superstring theory. Such a theory should be able to be captured in a short equation, not more than an inch or so long. He has accomplished it, and here it is:
Response: I have no idea what this equation says, and am in no way qualified to evaluate it. It represents the vibrations of really tiny strings in the 10^{th} and 11^{th} dimensions, and I understand that the high degree of difficulty in understanding it is not just limited to us amateurs who like to read books about math.
(b) The math for Kaku’s string theory is analogous to that of topology, much of which was first formulated towards the end of the nineteenth century. It is an extension of algebraic geometry, reaching into higher and higher dimensions. As Kaku tells it, those who contributed to the development of topology took great delight in the fact that this was math at its best, namely, math that would never have any practical applications since it includes multiple dimensions beyond our usual three (or four if you count time).
Response: The more dimensions you give yourself to work with, the more problems you can solve. If you only had two spatial dimensions, you could have a flat depiction of the eye of a needle and some thread, but you could never thread the needle. To do that, it takes three dimensions. Topology uses that principle and develops some truly bizarre models of algebraic objects in greater spatial dimensions. I’ll leave it to Professor Kaku to judge to what extent the mathematicians delighted in the socalled lack of applicability.
(c) Along came Michio Kaku, who used 10 and 11 dimensions in order to come up with his superstring theory. Mathematicians were surprised, if not shocked and appalled.
Response: As well they should be. You’ve got to be in awe of Kaku’s work.
(d) The vibration of these strings constitute the “music of the spheres,” and the very voice of God. He, Michio Kaku, has had a glimpse of the core of the universe, where he finds God communicating to him through the equation that he, Professor Kaku, has constructed.
Response: It would appear that Michio Kaku has endowed himself with the offices of prophet and priest in the religion of superstring physics. He has departed from physics and is trying to do theology, and the result is dubious.
We need to leave the more specific content of this religion for next time.
The “God” of some Scientists
In rereading some of my blog entries over the last 11 or so years (the anniversary will be July 3) , it seems to me that about a quarter of them begin with some kind apology for having skipped so many days, dragging out a series, or whatever. I have a feeling that, just by itself, that gets tiresome to read; I know it is for me. So, I’ll try to restrain myself. In addition to posting this entry, I do need to connect it to all of the previous ones of this series that began with the contemplation of phi (ϕ).
Yesterday, Wednesday, was a beautiful day with temperatures in the 70’s, sunshine, and a slight breeze. June and I both worked in the yard, June more than me. I didn’t get much done because I spent over an hour in a losing battle trying to get “weedeater” string fit properly onto a spool. It’s been a source of frustration for June, with her physical limitations due to her fibro and the weather oscillating between lots of heat and thunderstorms, that she hasn’t been able to work with her flowers as much as she wants to (and as the flowers want her to), I’m really glad that she had quite a bit of time for it yesterday, though today she has no physical reserves left. I continued my current practice of trying to go to the pool, and I raised my personal best at the pool to 10 lanes in a row.
On Monday, I went to Menard’s to buy a new door for the basement to the outside. That will be my major summer time project for the next couple of weeks: Take out the old basement door and replace it with the new one. The door is preframed, and so it sounds easypeasy, and you may think that a couple of weeks to install a door is awfully slow. Well, yes it is. But there are two reasons. 1) The frame has to be fastened to the inside of the door opening, i.e. into a larger frame. I think I will probably have to replace that one, too, but I won’t know for sure until I can take a good look at it with the old frame gone. Then I’ll know what, if any, lumber to buy and mount it. That’ll probably mean a couple more trips to the store, unless I find some suitable leftovers here in the garage. 2) I’m slow. June and I have discovered over the last few years that, if we try to start and finish a project in one day, it’ll probably take longer to get it done due to the inevitable crash and exhaustion than if we space out a project over a number of days.
In what follows, I’m going to refer to three videos, and—if at all possible—I’ll try to embed them here next time. They are about the concept of God, as espoused by Albert Einstein and Michio Kaku. There’s undoubtedly little or no need for me to introduce you to Albert Einstein. Michio Kaku is becoming quite well known as well as a theoretical physicist, popularizer of what is trendy in subatomic speculations, and the originator of string theory. He is also outspoken on his beliefs concerning God.
I’m obviously not going to provide a linebyline commentary on all of that’s being said. My choice of details is simply governed by what peaked my interest.
Let’s start with the video that is entitled “Is God a Mathematician”?
We need to interpret that question in such a way that it makes sense. Obviously, God does not do long division; he already knows the square roots of all prime numbers, and it will be of no news to him when (or if ever) we finally find out whether "all of the significant zeros of the zeta function have real part ½. " That last phrase is the celebrated Riemann hypothesis, and mathematicians have struggled for over a century now, trying to find either a proof or a disproof for it. In the meantime, God already knows the outcome; he just has chosen not to reveal it. (If he always immediately solved our puzzles for us, life would get awfully boring.) The real meaning of the question, as I would read it, is whether God intentionally created the universe with the mathematical orderliness and symmetry that mathematicians discover in it, including the interesting features of the numbers themselves.
Whoops! I just put myself into the position of having to add yet another level of stacking. But really, my patient readers, one cannot write about God’s creation of numbers without mentioning the view held Leopold Kronecker (18231891). It would be like writing about art museums in Paris and making no mention of the Louvre. I must begin by saying that Kronecker made numerous serious contributions to math and was by no means a crackpot. However, his philosophy on the nature of numbers did pervade his work, and the active way in which he promoted his views made for sour relationships with many of his colleagues. If he is at all known among sideline math fans, such as your bloggist, it is probably for the alleged statement:
"Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk."
The dear God made the whole numbers; everything else is the work of humans.
First, about the quote itself. It is frequently translated as “God made the integers,” in which case he could have been referring to the various numbers that classified as ℤ, which includes zero and negative numbers. I believe, though, that he was really restricting himself to ℕ, the positive integers. Regardless, there is not much sense in exegeting the quote since it has come down to us third hand as a report of what he had supposedly said to someone else in a personal conversation, and so we can hardly be sure of the exact wording. Still, the sentiment fits in well with his outlook on math.
Another, probably apocryphal, statement is associated with the occasion when a mathematician named Ferdinand von Lindemann (18521939) had published a significant proof with regard to the nature of π and received welldeserved acclaim. Kronecker supposedly asked him the rhetorical/polemical question: "Of what use is your beautiful proof, since π does not exist?"
Whether Kronecker actually made either of these utterances has been questioned. Nonetheless, they do represent his basic outlook on the reality of numbers. If you don’t recall the meaning of the 𝕙𝕠𝕝𝕝𝕠𝕨 letters, here’s a quick summary. If you totally missed that earlier episode, you can go back and read the descriptions in greater detail.
ℕ 
Natural numbers: The positive integers beginning with 1. 
ℤ 
Integers: adds zero and negative numbers. 
ℚ 
Rational numbers: adds fractions and closed decimals. 
ℝ 
Real numbers: adds “irrational numbers,” including ϕ, π, and √2. 
ℂ 
Complex numbers: adds √–1, called i. 
In Kronecker’s view, only the numbers that God created, ℕidentified with or maybe also ℤhave any reality. Thus, he excluded fractions (ℚ), the socalled “real numbers” (ℝ), and, quite obviously the complex numbers (ℂ), which are still designated with i, the "imaginary" square root of 1. π lives in ℝ, of course, along with his fellow celebrities, ϕ, e, √2, etc. Kronecker was convinced that ultimately the only real objects in math were these “whole numbers,” and theorems and proofs that went beyond that perimeter were useless. Thus he very strongly objected to many of the innovations put forward by some of the most recognized mathematicians of his day. For example, in his position as editor of an important journal, he refused to publish several papers by Georg Cantor, the father of set theory, who had concluded that the infinite set of real numbers (ℝ) was larger than the infinite set of natural numbers (ℕ).
Cantor's idea does sound just a little crazy when you first hear it, doesn’t it? And one can understand that Kronecker might have believed that Cantor had lost his common sense. Nevertheless, I would still have to go with Cantor, if my vote were to mean anything. Socalled common sense is not necessarily in a position to judge uncommon objects or situations. Regardless, Kronecker had little use for his sets. In our contemporary world of math, Kronecker has received an echo in the work of Doron Zeilberger of Rutgers University.
Approaching this point of view from the perspective of a philosopher, it seems to me that you can’t have one or the other. If God created the natural numbers ℕ, then he also created the rest of the number systems that stem from ℕ.
ℕ 
ℤ 
ℚ 
ℝ 
ℂ 
Comments 
4 + 5 = 9 




Within ℕ there’s no limit as to how many numbers you can add. 
4 – 5 = –1 ⇒ 
–1 [integers] 



But when you start subtracting you may need to leave ℕ and resort to ℤ. (“ℕ is not closed.”) 
4 – 4 = 0 ⇒ 
0 



ℕ may also require 0 as an integer, again needing help from ℤ. 

16 ÷ –4 = –4 



Within ℤ, more calculations are possible than in ℕ, e.g., using negative numbers 
4 ÷ 16 = 1/4 ⇒ 
⇒ 
1/4 [fractions] 


But ℤ is also not closed for division when a solution requires a fraction. Then we must take recourse to ℚ. 
2 × 2 = 4 
4 ÷ 2 = 2 
√4 = 2 


Everything seems to be going nicely in ℕ, ℤ, and ℚ when we let 4 and a couple of 2s play together. In ℚ we can describe 2 as the square root of 4, thus square roots have a legitimate place in ℚ, 


√2 = 1.4142 …, –1.4142 … ⇒ 
√2 [irrationals] 

But what worked so nicely for 4, does not work for 2. Now ℚ needs ℝ, the set of “real” numbers, which includes the irrational ones. 



√–2 ⇒ 
2i [complex] 
If –2 is a legitimate number from ℤ on, and it is “legal” to take square roots since ℚ, then we need to accommodate square roots of negative numbers. We do so by making use of i and the complex numbers. 
I trust that my point is clear and, hopefully, plausible. If you start with ℕ and proceed courageously to unpack what you have and what you can do with it, sooner or later you should wind up with ℂ. [See Robert and Ellen Kaplan, The Art of the Infinite (Oxford, 2003) for an extended rhapsody on this theme]. You can’t pick and choose. If God created the natural counting numbers of ℕ, then the implications and sound theorems that follow from basically logical operations on them are also a part of his creation. If fractions, let alone the real numbers, are merely human inventions that are not real, then the natural numbers that make them up can’t be real either. If 3 and 4 are real, dividing 3 by 4 does not make them unreal, and thus ¾ must may also partake of whatever reality numbers have.
Let me quickly add this. There are three major lines of understanding the reality of numbers:
1, The “Platonic” view, which holds that numbers have genuine reality. It would seem that Kronecker holds that view for the “whole numbers,” but only for them. Doron Zeilberger , whom I mentioned above as an echo of Kronecker, stated explicitly:
I am a [P]latonist, and I believe that finite integers, finite sets of integers, and all finite combinatorial structures have an existence of their own, regardless of humans (or computers). I also believe that symbols have an independent existence.
I am not sure if Zeilberger at the time of that presentation in 2001 was aware that Kurt Gödel, who becomes one of his foils, was a Platonist and that his “incompleteness theorem” was intended as a reductio ad absurdum of nonPlatonic views. (Please see my article with Michael Anderson, "The Strawman Strikes Back: When Gödel's Theorem is Misused"). Notsobytheway, in this context, the word “Platonic” somehow has become the convenient catchall term for the idea that numbers have their own existence, but one does not need to subscribe to Plato’s Forms to hold to this view.
Looks like discussing the videos will have to wait for the next entry.
Eight lanes back and forth at the pool Thursday. Only four on Friday, but I was pretty exhausted before I ever got into the water. On Thursday I had taken the red No. 14 Tony Stewart and A.J. Foyt Sears Craftsman Silberpfeil tractor out to cut the grass and wound up with a flat rear tire. Friday, I tried to fix it, but each stage—removing the wheel, jacking up the tractor, pumping air into the tire, etc., not necessarily in that order—came with its own subproblems to the point where I finally decided that on Monday I’ll take the wheel to Troy Greer’s and ask them to, please, fix it. I'm hoping that they won't just inform me that the rim is bent. Today (Saturday) the pool was too crowded to do a lot of uninterrupted laps.
In line with my normal practice, I’m not making a whole lot of progress on this series as a whole because I keep running across information that I find to be either necessary, helpful, or just plain cool. So, this post also does not go nearly as far as I wanted it to, but at least I have started to raise the theological questions.
The ubiquity of the Fibonacci numbers and their steady companion, ϕ, has become the occasion for much thought and writing about the mathematical regularity and apparent intentionality of the universe. In fact, the closer we look, the more astounding things become.
Again, because in this series I’m trying to stress the beauty of the numbers themselves, let me give you another example that strikes me as really astounding. We all know about ϕ’s big brother pi (π) mostly for his part in the formula for the area of a circle:
A =πr^{2}
But π also shows up in places where we would not expect to see him. Skip down if you like.
Let us look at an interesting problem, known as the "Basel Problem" (Derbyshire 63) because it was in Basel, Switzerland, that it was posed. The problem, as originally stated by Jakob Bernoulli (16551705), went as follows.
Here is a convergent series, which means that it eventually approaches the limit of a specific number rather than diverging to infinity:
As usual, we can come closer and closer to the number, but will never quite reach it. A precise enough approximation of the number is
1.64493406693 …
Are you getting tired of those three little dots ( … ), which indicate that you’ll never truly get to the end of calculating the formula’s value? Apparently Bernoulli was. In a publication on various related topics, he included this series and asked for input from other mathematicians about this question. Would anyone be able to state it in a form that did not lose itself in the forever of infinity, but in a “closed” formula that could be substituted for it?
Leonhard Euler (17071783), whose original home also happened to have been Basel, came up with a solution:
I don't think I'd be going out on a limb when I say that most of us are rather surprised to see pi popping up there.
Very quickly let's switch to a different topic, this time probability theory. Suppose you are asked to pick two numbers at random from a set of integers. What is the probability that they will not have any common divisors? E.g., 6 and 8 would share a divisor (2), but 4 and 9 would not. The probability of picking two numbers without a common divisor is:
The actual number is ~0.6079 or ~60.79%, but that’s obviously not what strikes us. This probability calculation turns out to be the reciprocal of Euler's solution to the Basel problem! The University of Illinois has set up an interactive webpage that illustrates this formula with some input from you.
The marvels go on and on.
Christians, Jews, and others who believe in a personal God look at these wonders of the universe and see in them the magnificent hand of its Creator. How could they not? I’m not offering that sentiment as a piece of apologetic on behalf of the createdness of the world or the existence of God, but as a report on what is (or should be) an obvious part of what theists believe. Or at least should. Ben Stein, whom I was privileged to count among my blog readers for a while, made the film “Expelled: No Intelligence Allowed” to publicize the fact that the scientific world on the whole (including atheists and purported Christians) has slammed the door shut on even asking the question of whether the cosmos manifests intelligent design. The entire movie is available on YouTube There are exceptions, such as William Dembski, who has not allowed his views to be dictated by the strong arm of a supposed scientific consensus, at the expense of his ongoing career. As Mr. Stein's movie displays, even atheists who allow the question to be raised may find their jobs terminated.
There is another group of scientists, however, including some of the brightest people of the last 100 years, who have not been afraid to express belief in God on the basis of the mathematical coherence of the universe, although they do not subscribe to a traditional understanding of God. The concept of God, as they have been espousing it, is not that of a personal Creator, and it doesn’t even really fit into any other traditional categories. It is not pantheism in which God and the universe are considered to be identical; let alone deism according to which God created the universe and then abandoned it to run according to the program he implanted in it. In some ways it’s reminiscent of the God of process theology who is persuading the world to follow his directives, from the cohesion of molecules on up. Still, there is far too much metaphysics associated with process theism to suit these celebrity believers. It appears to be the very regularity of the universe, the math behind the phenomena, so to speak, that constitutes God for them. Next time I’ll continue with these observations on Albert Einstein and Michio Kaku, and then return to phi.
Woops! I meant to go on talking about ϕ, but got caught up in the agriculture of Indiana by way of a remark on the weather. That's why I like to describe my blog as "mercurial." Longtime readers will know that there have been other times when I was surprised at what I wrote about rather than what I intended to write about.
For the last few days I had to go without a certain medication due to some snafus beyond my control, and I’m very glad to have it back. It was not good, and that's all I canor maybe want tosay about that.
A couple of cloudy, rainy days, but the temperatures still are definitely in the summer range. If you should drive, bike, or walk to the outskirts of Smalltown, USA, you will find yourself facing corn fields, and you might just hear the corn growing. The standard expectation is that the corn should be “knee high by the Fourth of July.” Many of the fields are knee high already, and will be close to “person high” by Independence Day, I should think. The fields definitely will dominate the view for the next few months.
It looks to me as though the timing worked out quite well for planting corn (aka maize, btw) this year. The big question is always when the fields will be dry enough for tractors and machinery not to sink in the mud. Around March or April, after the snow has melted and we’ve gotten our usual overdose of rain, many of the fields look like lakes. The farmer has to wait for the soil to dry out, and appearances can be deceiving in that regard. An early unusually warm spell can make the fields look dry, but that part may only be a crust with a lot of mud underneath. This is not helpful, for one thing because the farmer’s tractor might get stuck, and for another because such a crust also stands in the way of further evaporation of the water underneath the crust.
Around here, for corn to have enough time to reach full maturity, it needs to be “in” by “Decoration Day,” as I once overheard a farmer say.
Anyway, I’m pretty sure that everyone got their corn planted before Decoration Day. One must realize, of course, that there are different kinds of corn with different requirements. Overwhelmingly, the corn grown here is “horse corn” earmarked to become animal feed or silage. It lacks the appeal of the traditional “sweet corn” which is a different species altogether. Horse corn stays out in the field until it's all dried out. You wouldn't want to eat an ear of sweet corn in that condition.
Speaking of corn, let’s say that you want to watch a movie, and you think a bowl of popcorn would really go well with it. Great! Just be sure you have real popping corn. If you just take a handful of any old type of corn, put it in your microwave or other appliance, and stand by, ready with salt and butter, you may find that it doesn’t work that way. The kernels are not going to spring to life, and you can put the salt and butter back in the cupboard. Popcorn is a different kind of corn from the others mentioned so far. It is distinguished by the fact that, when the kernels have matured, they surround a tiny little drop of water. The heat brings that little bit of moisture to the boiling point, which increases the internal pressure, and the kernel, along with most of its colleagues in the bag, explodes into the white treat we’re all familiar with. So, popcorn is also among the agricultural items that farmers plant around here. About half an hour’s drive from here there is the little town of Van Buren, Indiana, population 864 as of 2010 (Compare to Smalltown, USA,’s population of 5,145. Both are shrinking.) It declares itself to be the Popcorn Capital of the World and celebrates an annual popcorn festival.
But, as another saying goes, “There’s more than corn in Indiana.” Beans (i.e. soy beans, though nobody call them that around here), go in about a week or two later than corn. There are also a few occasional wheat fields; it’s “winter wheat,” which means that it was planted way back in last October or so. You don’t see anything of it all winter, but these fields become some of the earliest and brightest displays of green when spring arrives. By now the wheat has attained its golden color, and it should be ready for harvesting pretty soon. The farmers are also getting ready for the first round of mowing hay.
As long as I’m pursuing this subject, you will also see occasional tomato fields in Indiana and surrounding states. According to the Wikipedia's statistics, if you’re in Indiana and you see a tomato field, there’s a 95% chance that the harvest will be processed into one of the many kinds of canned tomato products at the Red Gold plant, just a short three miles west of here in tiny Orestes, Indiana (pop. 414, not shrinking). Red Gold also has contracts with 80% of the tomato growers in the rest of the Midwest.
I might mention that, despite a strong heritage of sorghum (aka milo) production in Indiana, it is no longer a major crop here. The Hoosier State had been the nation’s leader in this regard at one time, but it became unprofitable. Nevertheless, there still are a number of sorghum fields, even if the US department of agriculture doesn’t acknowledge their existence. In its early stages, corn and sorghum look a lot alike; the distinction being that sorghum plants are a little shorter than corn and its leaves are spikier. Later on, though, sorghum clearly stands out with much larger and wider tassels than you’ll see on corn.
Pictures courtesy of Sorghum: The Smart ChoiceAll About Sorghum 
Some efforts are being made to reintroduce sorghum as a profitable crop on both public and private levels.
Next time, if I can keep my mind from running off into different directions, the perennial question: Is God a Mathematician?
Most of this was written originally on Saturday evening, but needed some refinement and a lot of uploading of formulas, plus adding enough cute stuff to give it appeal to a general public. Hope you like the Staypuft Marshmallow man gif I concocted.
The weather continuous to be quite nice, and definitely appropriate for summer. We are heading into the high nineties again today. A couple of weeks ago I procured my annual season pass to the Beulah Park Pool, here in Smalltown, USA, and I’m making good use of it, every day if I at all possible. It’s funny (so to speak): on the first day I went out there this year, I felt like I had lost an enormous amount of strength in my arms and legs, and I wasn’t surprised. I couldn’t even manage to swim one entire lane without taking a break. However, by this past Friday, after only about a week, I managed four, and as of yesterday and today, I’m up to six already. That’s definitely also been a surprise, and obviously a good one, for which I’m grateful. On most days, after swimming one set, I usually take a break and visit the water slide once or twice to satisfy my need for speed. Then I do another set, casually and without trying to go to any limit.
Well, let’s get back to the numbers.
There are several mathematical surprises connected to the Fibonacci series in its own right, apart from its involvement with ϕ. For example, even though ϕ is definitely an irrational number, it does manifest a certain amount of regularity, such as certain digits recurring at precise intervals. As intriguing as those things are, I need to refer you to Livio for details. If I don’t observe some limits, this series would still be going a year from now, and—who knows—even the spammers might be scared away by then. So I’m going to continue to focus on ϕ as much as I can, and talk some more specifically about the relationship between the Fibonacci series and ϕ.
I’m going to introduce a new feature in this entry. The post would become meaningless if you or I just skipped all of the details, but I think I’ll keep more readers interested if I show them what they can skip without missing out on all of the content. So, I will introduce the problem for the day, but then give you the option of clicking on a link that will take you right to the conclusion without going through all of the steps to take you there. You will see it shortly below as a green rectangle.
There is a formula known as “Binet’s Equation,” named after Jacques Phillipe Binet (17861856), who was actually not the first person to discover it, but apparently created more interest in it than other famous mathematicians before him. Now, I’m going to trot out Binet’s equation, and you may just be horrified when you first see it. It looks like a huge monster, perhaps reminiscent of one such as the StayPuft Marshmallow Man ® from “Ghost Busters.” We’re going to describe Binet’s equation, and pare it down until it becomes one simple calculation. Then we’ll use it to find out what the 12th Fibonacci number is.
[Added on Monday: When I was working on this on Saturday night I had no guarantee that it would actually work. To be more specific, I assumed that the formula was right, but I did not know if I was interpreting and using it correctly. If it didn’t work out, I had no one to ask, at least immediately, what I had done wrong. Thus, my emotional reaction at the end was totally authentic. I couldn’t believe that I had brought of both stages: understanding the formula (a good first step to clarifying it for my readers) and applying it in order to get the right answer. ]
Here is the promised link that allows you to skip over the material that looks intimidating.
Here goes. This is Binet’s Equation:
Intimidated? You should be—but only until we’ve taken it apart and seen the simplicity that’s behind the apparent complexity.
Seriously, I’m going to go extremely slowly in clarifying this equation and highlighting all steps. Maybe you will realize that working with such formulas one does have to be careful, but it does not have to be “*cough* laborious *cough*, if I may tease John of LRT one more time (all in good fun, I hope he understands if he should ever read it). One of the nicest compliments I ever received on my writing was with regard to my commentary on 1 and 2 Chronicles, when the editor at B&H told me: “Win, you have made Chronicles come alive.” I don’t know if it’s a comparable challenge, but I shall try to make Binet’s formula come alive as well.
“Fn” stands for which Fibonacci number in the chain you wish to calculate, i.e. the “nth” number. This is totally of your own choosing. For now, we’ll try to use the formula to figure out the 12th Fibonacci number, a number we can easily verify by just adding up the Fibonaccis. Maybe later, then, we’ll try our hands at calculating a much larger number, perhaps F95, but first we’ll have to see if we make it through this trial.
The expression “for n ϵ ℤ” reminds us that n has to be an integers. The Fibonacci numbers per se are not irrational, and are inhabitants of ℤ. The irrationality of phi and squareroot of 5 comes into play insofar our answer may need rounding up or down to keep the solution within the ℤ circle as well. From here on out, we're going to dispense with that little tail.
Wow! What are we going to do with that messy clutter highlighted in red? Actually, you may know already what we’re looking at here once I let it stand by itself. It’s the positive root of our good old equation for ϕ:
x^{2}x1=0.
namely
So, we can substitute ϕ in for that formula, and life is beginning to look a lot simpler already. We now know that, when the time comes for actual calculations, we can insert 1.61803 as the value of ϕ.
A similar thing is true for the next packet that I’ve highlighted in red.
This is the negative solution for phi’s equation, and you may or may not recall that it’s value is the same as the negative conjugate of ϕ, but terminology aside, you may remember that it amounts to 0.61803. Rather than writing “1/ϕ”, which would mean that we're building up fractions again, I’ll use an expression that’s equivalent, but a little easier on the eyes, ϕ^{1.}
Let’s not forget that, when we eventually subtract that negative number, we can substitute a “plus” sign for the two negatives—if we ever have to do so. The last thing I can point out before we actually install some numbers is that both ϕ and its negative reciprocal will be raised to the power of n.
Now we’re looking at the formula that we want to instantiate.
I know; I know. You’re probably still not buying into my claim towards simplicity. Okay, I have one more trick up my sleeve to try to persuade you, and it's a good one. There’s a corner we may cut without doing ourselves any serious harm (just don’t run with the scissors, please). That whole –(ϕn) business is going to get extremely tiny, yea infinitesimal, so quickly that it can be pretty much treated as negligible for our practical purpose. I’m allowed to say that because this formula does have a practical purpose, namely to find a specific number in the Fibonacci series. (Alright, different definitions of “practical” perhaps.) If we give ourselves permission to leave off the negative part, we can actually calculate the “nth” Fibonacci number of the series with the simple formula: One multiplication (ϕ raised to the nth power) followed by one division, dividing the product by the square root of five.
We set out to find the 12th Fibonacci number, and now we’re at the point where we can put some numbers into that simplified formula. If our result comes out really skewed we’ll have to work a little harder. Let’s substitute 12 for n:
If you’re using a sophisticated program, you can just plug in the symbols. I don’t expect that too many of us carry a value for ϕ^{12} in our heads, but Wolfram/Alpha has given me an answer of 321.987… . We can use a value of 2.236 for √5. So we can take that number and divide it by the number that Wolfram has given us. The crucial moment is drawing nigh:
We do the division and get …
Let me tell you how nervous I am right now. As I said above, I have not done this before with all of the appropriate details. Before starting to write down anything formal, I gave it a rough onceover trying just to see if it might work, but now I’m really wondering. I’m sure you’re in suspense, too, about whether I worked it out so as to get a plausible result. Of course, it’s possible that it didn’t. I am fallible and prone to small mistakes that generate drastic changes in the outcome. That reminds me of a time when …
Okay, okay, I’ll stop dawdling. It’s taking me a bit to get past this approach/avoidance dilemma. Let’s take it to Wolfram then and have it do the last division for us. The expected answer was 144. Here is the result of applying the truncated form of Binet’s equation:
144.0015 …
I can’t believe it. I really, honestly did not expect anything nearly as close to 144. I’m stunned and almost emotional. This was a long and winding road. Thank you to those of you who stayed with me through the entire length of this entry. I know it was demanding. But isn’t that result a whole can of highoctane awesomesauce‼ Back in my childhood days in Gymnasium, the math teacher, Frau Dr. von Borke, had us memorize the multiplication table for up to 20. Thus I realize that 144 is not only the 12th Fibonacci number, but also 12^{2}. We can’t derive a rule from that fact, but we can see another piece of that beautiful mosaic of numbers. When God built in numbers onto the universe he created, he not only gave it regularity, but he infused into it a beauty that we ignore to our own loss.
It’s too bad that it has become almost fashionable these days to promote yourself as someone who doesn’t get along with numbers, wearing that selfdeprivation almost like a badge of machismo. Look at what you’re missing!
I wish all of my readers a day filled with beauty on many different levels.
Thanks to everyone who tried the Tripod sites. They are working now. I still can't get into my file manager, but for the moment there are lots of ways of getting around that, and I shall be patient. Come to think of it, being patient is about all I can do. I can wait with a happy face or with a frowning face, but wait I must.
For some unknown reason Tripod/Lycos has been inaccessible for the last several days. I house most of my online pictures with them, as well the majority of my subdirectorysites, including the presently growing series on PHI and the allencompassing www.wincorduan.com, I also cannot get to their administrative sites, and my emails seem to vanish in the ether. From what I've been able to test, the problem lies with neither my computer nor any browser. I have no idea what's going on with them, and I hope that the issue will be fixed pretty quickly.
In the meantime, a lot of material is missing, such as quite a few pictures in this blog. I will try to transfer to Bravenet as much a I can as soon as I can; much of it is stored in my "local" files that I can reupload, but some things are in their appropriate form only on Tripod. Obviously, I can't transfer those until I get access to them. I guess that if I do get access to them again, I won't need to transfer them, but it may not be a bad idea to have important thinks saved in several places.
So, please, be patient with me about the absence of certain illustrations and some resource sites. And, if you happen to think of it, would you please from time to time check either one of these addresses, both of which will send you to slightly different versions of my home page:
www.wincorduan.com or win_corduan.tripod.com,
and let me know if you actually got through.
I will do the same of course, but someone else may be the first to find that we're hooked up with Tripod again.
Mr. Lycos or Mr. Tripod (or whatever your names are), would you please 1) acknowledge my existence, 2) fix this mess, 3) show me what I can doif anythingto help fix this mess, and 4) give me a prorated refund for the sizable amount of money I spend on you.
Thanks, everyone!
Examples of the occurrence of the Fibonacci series in nature abound, and so do their descriptions. Therefore, apart from referring you to some websites and simply mentioning a few examples, I will not focus on that aspect. It’s been done often and it’s been done well. It’s the mathematical relationships that I’m trying to clarify.
“John” (no further name seems to be available) wrote an interesting article for the website, Let’s Reason Together, “It’s All in the Numbers.” He begins with the confession:
I’m not a mathematician. In fact, I generally have an aversion to numbers. They’re restrictive, require systematic, stepbystep (*cough* laborious *cough*) methods to manipulate, are terribly predictable, and generally unresponsive to creativity—at least, the rightbrained sort (my sort) of creativity. 
I’m taking what “John” said there as a somewhat humorous expression of his humility. Part of what I’m trying to show here is that, even though working with numbers may require systematic step by step analysis, doing so properly also needs “rightbrain” creativity. Furthermore, its laboriousness is not any more tedious than what I see many serious artists practice on a daily basis. Producing a work of beauty may necessitate long hours of intense labor. Take the Taj Mahal, for instance … Furthermore, if one wants to write on a phenomenon in math, one should ideally be on good terms with the subject, and I’m sure that most people would agree with that sentiment.
“John’s” article mentions four areas of nature in which the Fibonacci numbers play an organizing role:
1. the sections of pine cones and pineapples, which are arranged in a Fibonacci spiral;
2. the spiral arrangement of seeds within a sunflower;
3. the genealogical pattern of the male population within a beehive;
4. the relationship of the phalanges and metacarpal bones of the human hand.
We can add other examples, such as
5. the number of petals in a rose.
These are realities that Christians can point to and say that there must have been some intelligence involved in constituting the world in this marvelous way. Sure, we can find reasons why these arrangements are in the pattern of the Fibonacci series insofar as they contribute to the wellbeing of a plant, an animal, a geological formation, etc., but that observation only strengthens the argument that these examples are not random aesthetic or mechanical coincidences. If they serve a purpose, then there’s all the more reason to believe in an intelligent Creator behind the construction of the universe.
Sadly, once again I need to recite my litany about wishing that some Christian apologists were a little more careful when they assemble their evidence on behalf of God and the Bible. For example, the article by Fred Willson of the Institute for Creation Research, “Shapes, Numbers, Patterns, and the Divine Proportion in God’s Creation” [Impact: Vital Articles on Science/Creation,” #354 (December, 2002)], quite uncritically compiles what appears to be every claim for the golden ratio ever made, including some that are dubious at best, thereby undermining the credibility of what could have been a good case. E.g., his very first example consists of a correct description of the chambered nautilus, which he then labels, incorrectly, as displaying the “golden spiral.” (Note my earlier point that the mollusk in question does manifest a logarithmic spiral, but it does not fit the golden ratio as usually understood. At the time I just referred to this error in general; I had not yet seen Willson’s article. )
I trust that John of CLR will indulge me as I return to indulge myself in the mathematical aspects of the Fibonacci numbers and the number phi. Theoretically, there need not have been a Fibonacci series in order for there to be a phi, since, as we have shown, the actual birth of phi occurred in geometry and number theory. The Fibonacci series can be described with an equation, in which Fn stands for the rank of a given Fibonacci number. What is the “nth” Fibonacci number?
(Fn = Fn1 + Fn2)
e.g., the 9^{th} Fibonacci number = the 8^{th} Fibonacci number plus the 7^{th} Fibonacci number
So, we can just remember the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, … and fill in the slots.
34 = 21 + 13
This procedure is indeed lame, predictiable, and extremely slim on creativity. The matter of far greater interest is, of course, that the further we go, the closer the ratio of one number to its predecessor converges to phi. I realize that I have heretofore spoken of this relationship between Fibonacci and phi in a somewhat deprecatory manner, but, since you now realize that phi is not directly derived from the Fibonacci series, I can go back and endow the idea of convergence with the respect that it actually deserves. By “convergence” we refer to the properties of an ongoing series of numbbers whose numerical value comes increasingly closer to a particular number, ultimately coming so close to that number that the difference appears trivial.
John Derbyshire [Prime Obsession (Washington, DC: Joseph Henry Press, 2003)] presents us with a good illustration of convergence by contrasting it with its opposite result, called divergence. Most of this entry from this point on summarizes and maybe clarifies (if necessary) Derbyshire’s exposition. “Divergence” means that the value of the series keeps growing and, thus, eventually could be said to be infinity (∞). He illustrates divergence by using the socalled harmonic series, which consists of the reciprocals of regular counting numbers (those that inhabit ℤ).
A close relative of the harmonic series does, in fact, converge. It’s as simple as recreating the harmonic series, but using exponential powers of 2 as the denominators:
For a moment there you might think that, similar to the harmonic series, this one will also extend to infinity, but it doesn’t work out that way. To be sure, the number does keep growing; however, the rate of growth declines rapidly so that, as you go further along the best you can get is nearidentity with a finite number. Let’s see what happens when we add it up this far:
That’s pretty close to 2, and the further we go, the closer we will get, though we’ll never truly reach it. (Is anyone else reminded of the sizes of wrenches and sockets in an Americanstyle set of tools?) We can say that the formula approaches 2, and that’s good enough for many purposes.
Similarly, the ongoing ratio of the Fibonacci series approaches phi.
and that, too, is good enough for many purposes.
One such purpose is to help us determine the “nth” Fibonacci number when it’s a long way up the chain, say the 95^{th}, without having to memorize the entire set of Fibonacci numbers or go through the tedious and (*cough*) laborious (*cough*) method of generating the lengthy chain of 95 links. But the clock, my energy, and your patience all lead me to realize that I better save that excitement for the next entry.
(Also, this is the first time that I’ve availed myself of MSWord’s equation feature, which is pretty nifty. However, I wonder to what extent these formulas are going to translate into html. If I thought that thumb crossing had any value, I’d ask you to figure out how exactly one crosses one’s thumbs and then maintain your digits in that position.  Alas, it didn't work, so I needed to convert the equations to pictures for now.)
Well, I've played my little game long enough, and I thank you for your patience. But now the time has come to start counting lagomorphs.
I hope that in my preceding discussion I have made it clear that the socalled golden ratio number φ (phi), has a life of its own, apart from the Fibonacci series. If you have caught on to that point, we are now ready to talk about the Fibonacci numbers and how they are intertwined with phi. (If you still say that phi is derived from the Fibonaccis, you're probably only teasing me, or you didn't read the preceding posts on the topic.)
Leonardo Bonacci (1170?1250?) made a number of important contributions to math, the most significant of which is undoubtedly his promotion of the Indian/Arabic numerals in Europe, which made life a lot easier to anyone having to undertake any calculations. If you've heard of Leonardo at all, it was probably under his nickname, Fibonacci, and most likely in connection with the series of numbers that is designated after him.
The series is embedded in a puzzle that Fibonacci posed to his reading audience (Livio, 96):
A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive? 
Please follow along with the picture as I walk us through the beginning of solving this puzzle. I might clarify, by the way, that we are counting the mature pairs only, just because that's how it's presented. The outcome would be no different if we counted immature pairs alongside the mature ones, except we would have only one numeral 1 at the outset, rather than two of them.
I trust that you can see the pattern that is developing. Month by month the collection of rabbits is increasing. The numbers of mature pairs that we have now for the first six months are:
1 1 2 3 5 8
Each new number is created by taking the last two and adding them. So, let's continue this pattern until we reach the twelfth month, which will give us the answer to the specific puzzle that Fibonacci posed:
1 1 2 3 4 8 13 21 34 55
So, there will be 55 pairs of rabbits (110 individuals) living in the man's enclosure after a year. That doesn't sound too bad, but keep in mind that we are now entering a phase when the numbers will grow quite rapidly. Give it another three months, and the number will be 233 pairs, which is to say, of course, 466 grown rabbits. Then there will also be another 144 immature pairs (288 young rabbits), giving us a total of 754 rabbit hopping around, looking for lettuce and carrots.
As fans of the Fibonacci series know, and as I have intimated already, there is a close relationship between the Fibonacci numbers and φ. Take a number in the series and divided by the previous one, and you'll get some number that is in the vicinity of phi. So, let's look at two of such ratios.
1) The number of the twelfth month (55), divided by that of the eleventh (34): 55/34 = 1.617647059 ...
2) Since I brought up the fifteenth month, let's see what we get when we divide 233 by 144. The answer is 1.618055556 ...
Let us recall the approximate numerical value of phi: 1.618033988 ... . We can see that the longer we go on with the process, the closer we will get to phi. But there's something else that may not jump out at you immediately because of the many digits. Let us put these three numbers into numerical order, lower to higher:
1.617647059  1.618033988  1.618055556
The ratio of the 12th number to the 11th was below phi, whereas the estimate of the 15th to the 14th was a bit higher. I knew that would be the case, even before I tested it, just to make double sure. This is one of the interesting aspects of the relation of the Fibonacci series to φ: the different ratios of one number to its predecessor will always be an "approximation," though they get extremely close. But the direction of the error will alternate. The ratios with even numbers will be slightly lower than an more accurate rendering of φ, while the odd ones will exceed it just a little. Needless to say, as you have seen for yourself if you did the little exercise at the outset of this series, even with nonFibonacci numbers the approximation gets very close, how much more within the Fibonacci series!
Next time: some more cool stuff about the Fib. numbers, as well as some theologial reflections.
Yesterday was a gorgeous day in Hoosierland! Blue sky, sun shine, no need for an air conditioner at the moment. Today not so much. It’s supposed to get very hot again, so I’m glad we were able to make the arrangements to fund a new system. D. has already removed the old unit, will purchase the new machinery tomorrow on our behalf, and have it in place, hopefully, by the end of the week.  Well, today that didn't happen. The funding is not yet showing up in our bank account. Hopefully it will be there tomorrow, so we can get the show on the road.
I’m going to continue with my discussion of phi. If this is the first time you happen upon my blog in a while, you may be a little confused as to the topic, particularly coming into the middle of an ongoing series. So, please know that the first four entries, which go all the way back to last year, are collated into a single document, to which I will add this one and the last two as soon as I can, so that it’s all in one place, and you don’t have to find your way backwards through my blog archive in order to read what came before.
Allow me to revisit what we’re doing here. We’re examining the nature of phi (1.61803…), the golden ratio constant, and for the moment we’re doing so as far away as we can get from the Fibonacci series and its exemplifications in art or nature. My point is to try to demonstrate its inherent elegance, which stems from its Creator just as much as the arrangement of flower petals and other “fibonaccied” items in nature.
Let’s go back to the question of two days ago: How contrived is the golden ratio? For example, take the golden triangle, which includes phi as an important ratio. Is it something that has been concocted just to show off phi, or is the golden triangle something that we can uncover and discover in other places. A partial response has been that it is included in the formation of a regular pentagon, and, thus, can be discovered without being guilty of fabricating an artificial instantiation just to smuggle in phi.
Phi is also found apart from geometry in number theory, which we can combine with a little algebra. Remember that phi is just one member of the infinity of all real numbers, which, as demonstrated by Georg Cantor, has turned out to be larger than some other infinities. However, it stands out from this uncountable crowd, right next to p, e, i, √2, and a few others, due to its special properties or notoriety.
Here are two ways of finding phi without geometry. We will take a look at a couple of somewhat unusuallooking formulas and turn them into equations, which will resolve into phi. Here is the key: We came up with the formula for phi by setting the length of the base of a golden triangle as 1, and one of its sides as x. We were able to manipulate those numbers into a quadratic equation
x^{2} x 1 = 0
The method for finding phi tonight is going to consist of finding several equations that can also be rearranged into the same quadratic equations. I found the following two equations in Livio.
Let’s start with a formula that consists of an endless fusion of square roots. Given:
By the way, this thing is a formula, not an equation. You can only have an equation if there are two (or more things) that are considered to be equal, as, e.g., in
It would be extremely tedious to work out a value for x from this equation if we want to continue to add the square roots of square roots, etc. But there is an easier way to do so.
We can square both sides. Squaring x gives us x^{2}, and squaring the formula to the right resolves the first square root into a 1.
Now it is apparent that the collection of square roots that follows after “1 +” is still identical to the original given formula since both extend to infinity,
and, what’s more, we have already designated it as x above.
Then, substituting x for that nest of square roots, we get
which we can reformulate to fit the pattern we had looked for as
This is, of course, the formula which has phi as one of its solutions. Just think: we have derived it from that unwieldy collection of square roots.
So, now you feel like the Lone Ranger and want to save the world from more confusing formulas. That’s great. I’m with you all the way. Let’s try this monstrosity, which in mathematical argot is counted among a large group of “continuing fractions.” We start with a 1 and add the fraction, which has 1 for its first numerator and is followed by a neverending, alwaysrepeating denominator. Given:
Again, before we do anything else we need to turn this formula into an equation and label it with the variable x.
As you seek something wonderful in this equation, may I call your attention to the section underneath the topmost numerator of 1.
As in the previous equation, what you find there is actually identical to the originally given formula. There is no difference because both of them can be extended to infinity. And thus, we can safely apply the same letter variable, x to this new continuing fraction, which is also the old one. Remember now, that the area we have marked out is the denominator, and that the numerator of 1 still stands as before.
So, once we have substituted x for all of that clumsy denominator, we get a refreshingly simple equation.
Let’s get rid of the fraction 1/x by multiplying every term by x,
Then we have
and once more we can rearrange this equation into our favorite configuration:
Once again, we have turned a beast (the continuing fraction) into a beauty (phi). We have stumbled on yet another way of deriving phi without getting out our rulers or measuring tapes.
I’ll never be able to maintain a regular blog as well as take care of other things if I keep making myself include all kinds of diagrams and drawings. Still, it is fun, and for most of us a picture paints a thousand words—which is not to say that people who draw a picture necessarily forego an additional thousand words (ca. 1050 for this post). I started this entry yesterday (Monday), while still in Pokagon State Park. It was a good day, too. I had my first trail ride on horseback of the year, and June and I spent lots of time outside and swimming in the pool. Now we just got back to Smalltown, USA, and we need to see what we can do to get the new a/c system installed. Also, of course, I’m trying to finish this entry.
Before returning to phi allow me to mention that my next StreetJelly date will be my regular set on Thursday evening at 9 pm EDT. I had to cancel last week due to circumstances that weren’t entirely beyond my control. After all, I was not forced to go to the dentist. Regardless, I see no impedance for this week. I’m planning on doing a show of some of my original songs. That’s at StreetJelly.com.
Please let me remind you that all of the earlier discussions on phi (except the last one—for now) are collected in a single site so that you can read the previous sections in a sequence that makes sense.
I left off last night by showing that phi manifests itself when you bisect two adjacent angles of a pentagon and, thereby, create a “golden triangle.” Then we were able to give birth to more and more golden triangles of diminishing size by bisecting one of their base angles each time. There is also a process that gives rise to new generations of golden rectangles, as we shall see below.
By now I’m sure you have figured out why this rectangle should be golden: the ratio of the longer side AB to the shorter side AD is the same as the combined sides AB+AD to the larger side AB.
Now, we can lop of a square of length AD from the one of the ends rectangle, and we have a smaller rectangle left. I have placed the square on the left side of the rectangle. There is no rule governing that placement, nor can there be, since one can always flip the figure without doing it any damage. I’m placing my squares so that I can use the ongoing generation of golden rectangles to make a specific point in a short while.
The remaining rectangle (EBCF) now has the golden proportions. Let’s continue the process and remove another square designated by EBHG, and we have produced yet another golden rectangle, answering to the name of GHFC.
Are we done now? Only if you want to be. We can remove another square and enjoy the sight of golden rectangle GIJF.
And let’s do one more and call it GILK.
And so forth … This is another unique treat that phi brings to us: We can go on and on bringing out golden rectangles by removing squares from one of its side.
Let us now reverse this process by starting out with the smallest golden rectangle and adding squares to it so as to create a newer, larger one, which will yield another golden rectangle by means of the same procedure.
It is at this point that the placement of the square takes on significance. If I were to continue the enlargement procedure indefinitely according to my pattern, my arrangement will give us a spiral. In order to turn the tiniest of our rectangles into the next largest size, we’ll put a square underneath it. To reach the next size, we can place our square to its right. Moving on the next larger one, we can place the square on top of the one we have. Finally, to reach the largest size with which we began, we can expand it by means of a square on the left. Again, there is no point at which we have to stop, except for intrusions into our mathematical world, such as lack of available bandwidth, old age, or boredom. What you see is the beginning of a spiral. If we were to continue the process, the sequence can continue with the pattern of adding squares: down, right, up, left. Each time we get a new rectangle, it’s a golden one, and each one stands in proportion to all of the other by multiples of phi, and the spiral, that we call a "Golden Spiral" grows.
Here is a golden spiral from Wolfram Alpha. I chopped it up and turned it into an animation.
Are we still in math or, more specifically, geometry? Yes, we are. However, having shown you these constructions, I cannot forego mentioning one example of where we some people believe that they can see such a building process in action in nature, viz. an increase by a factor of phi in each particular stage.
The item in question is an animal belonging to a group of mollusks called "cephalopods" [“feet on the head”], known among his friends and family (e.g. the octopus) as the chambered nautilus. This animal sets out its life in a rather small shell, but as it grows older and bigger, it manufactures larger chambers for its comfort and convenience—modern living for mollusks! A common belief is that this growth occurs according to the golden ratio. Unfortunately for ardent enthusiasts of the golden ratio, this is not so. It’s really a shame because once upon a time even your bloggist, who holds a B.S. in zoology, had been a victim of the same misconception. Gary B. Meisner, who calls himself the “phiguy” and maintains a quite sizable website devoted to the golden ratio, explores a number of ways in which phi could be found as part of the proportions in the chambered nautilus, and leaves us with the ambivalent answer that, as yet, no nautilus that fits the pattern has been discovered, though it could not be ruled out that it will. For my purposes (and Meisner’s, if I read him correctly), that’s not enough to marvel at the golden ratio in nautili, though it leaves plenty of other things about which to marvel.
There are other, much better, examples in nature, but I’m once again postponing writing about those. There’s still too much I want to share with you about phi itself and where it shows up in its own world, the cosmos of numbers and formulas.